THE UN IV ER SIT Y OF MI CHI GAN COLLEGE OF ENGINEERING Department of Mechanical Engineering Heat Transfer Laboratory Technical Report No. 5 FINITE DIFFERENCE CALCULATION OF PRESSURE RISE IN SATURN S-IVB FUEL TANK Herman Merte, Jr. Chai Chin Suh Edward R. Lady John A. Clark ORA Project 07461 under contract with: NATIONAL AERONAUTICS AND SPACE ADMINISTRATION GEORGE C. MARSHALL SPACE FLIGHT CENTER CONTRACT NO, NAS-8-20228 HUNTSVILLE, ALABAMA administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR April 1969

TABLE OF CONTENTS Page LIST OF TABLES v LIST OF FIGURES vi NOMENCLATURE vii ABSTRACT x I. INTRODUCTION 1 II FORMULATION 4 A. General Formulation 4 1. Tank side wall 4 2. Liquid region 6 3. Vapor region 8 4. Interfacial heat and mass transfer 12 B. Transformation of the Partial Differential Equations 13 1. Liquid. region 13 2. Vapor region 14 C. Dimensionless Form of the Equations 20 1. Tank side wall 20 2. Liquid region 21 35 Vapor region 23 III. METHOD OF SOLUTION 26 A. Finite-Difference Forms 26 B. Stability Criteria 30 C. Computational Procedures 30 D. Computer Program 34 IV. RESULTS 35 A. General Assumptions 35 B. Variations Possible in Program 36 C. Input Parameters 36 1. Geometry 37 2. Heat flux 38 35 Acceleration 40 4. Fluid properties 40 5~ Miscellaneous 41 111

TABLE OF CONTENTS (Concluded) Page D. Results of Computations 42 1. Typical plots of streamlines and isotherms 42 2. Influence of change of grid size 51 3- Influence of heat flux distribution 61 4. Influence of wall material 64 5. Influence of body forces 64 6. Influence of ullage fraction 67 E. Discussion 67 APPENDIX A: COMPUTER PROGRAM FLOW CHART 75 APPENDIX B: LH2 COMPUTER PROGRAM LISTING 79 APPENDIX C: LH2 PROGRAM NOMENCLATURE 93 APPENDIX D: LH2 PROGRAM-DATA INPUTS 101 APPENDIX E: LH2 PROGRAM-TYPICAL OUTPUT (RUN B-H 47) 105 REFERENCES 119 iv

LIST OF TABLES Table Page I. Comparison of Geometry for LH2 Tanks 37 II. Tank Wall Properties 38 III. S-IVB Fuel Tank Heating Rates 39 IV. Computer Model Heat Flux Distribution 40 V. Index of Computer Runs 43 VI. Effect of Axial Grid Size 61 VII. Comparison of Heat Capacities of Tank and ContentsComputer Model 63 VIII. Comparisons Between Actual and Computer Model Heat Fluxes 70

LIST OF FIGURES Figure Page 1. Container configuration and coordinate system. 5 2. Control volume of vapor region. 11 3. Dimensionless isotherms-Run B-H 47. 45 4. Dimensionless stream functions-Run B-H 47. 53 5. Streamlines —Run B-H 47. 59 6. Effect of heat flux distribution on pressure rise. 62 7. Effect of wall material on pressure rise. 65 8. Effect of body force on pressure rise. 66 9. Effect of ullage fraction on pressure rise. 68 10. Composite computed pressure rise. 69 11. Effect of nucleate boiling on pressure rise. 71 12. Effect of nucleate boiling on local liquid-vapor interface mass transfero 73 vi

NOMENCLATURE A - Area, ft2 a - Cylindrical tank radius, ft; acceleration, ft/sec2 b - Total tank height, ft C,C - Specific heats Btu/lbm-~F p v Df- Substantial derivative of function f Dt e - Specific internal energy, Btu/lbm E - Total internal energy, Btu g - Acceleration due to local gravity, ft/sec2 h - Heat transfer Coefficient, Btu/hr-ft2- F; enthalpy, Btu/lbm h_ - Latent heat, Btu/lbm ~fg k - Thermal conductivity, Btu/hr-ft-~F P - Pressure, psia q - Heat flux, Btu/hr-ft2 r - Radial distance, ft Ar - Finite difference of radial distance, ft lbf-ft R - Gas constant, ibm- R t - Time, hr T - Temperature, ~R AT =T -T ~R w w sat AT - Specified. maximum value of AT, OR wmax w u - Axial component of velocity, ft/sec vii

NOMENCLATURE (Continued) U - Dimensionless axial velocity v - Radial component of velocity, ft/sec V - Volume, ft3; dimensionless radial velocity w - Mass flow rate, lbm/sec x - Axial distance, ft X - Liquid height in container, ft Ax - Finite-difference of vertical distance, ft Z - Compressibility factor, [1] Greek Letters - Thermal diffusivity, ft2/hr - Coefficient of thermal expansion, ~R-l y - Ratio of specific heats Cp/C, [1] E - Tank wall thickness, ft - Dimensionless radius, [1] An- Finite difference of dimensionless radius e - Dimensionless temperature: - Dimensionless axial distance, [1] A: - Finite difference of dimensionless axial distance 4 - Dimensionless stream function,'" - Liquid and vapor stream functions X - Dimensionless time AT - Finite difference of dimensionless time viii

NOMENCLATURE (Concluded) [ - Kinematic viscosity ft2/hr v - Dynamic viscosity, lbm/ft-hr p - Density, lbm/ft3 a' - Vorticity, [ft-sec] a - Dimensionless vorticity Subscripts d - Discharge f - Liquid or fluid g - Gas or vapor i - Liquid-vapor interface ~ - Liquid o - Initial state p - Pressurization r - Reduced s - Saturation w - Wall ix

ABSTRACT The pressure rise of a two-phase system in a closed container subject to an external heat flux is related directly to the temperature of the liquidvapor interface, which in turn depends on the heat and. mass transfer interactions between the liquid, vapor, and container. The problem is formulated here for a cylindrical tank with an axial body force and a symmetrically imposed external heat flux in terms of the transport equations. The temperature and velocity distributions are determined using a finite-difference method., which is coupled with an integral form of the energy equation to determine the pressure rise. The procedure adopted. takes into account the possibility of incipient and. nucleate boiling. Numerical computations are carried. out for liquid hydrogen in a large, partially filled tank under low gravity. This system models the one orbital experiment conducted to date which provides the only available experimental data. These are the data on pressure rise and system temperatures telemetered. from a Saturn LH2 tank orbiting the earth, the AS-203 low gravity orbital experiment. A discussion of the modeling is included. The numerical computations are carried out for various distributions of heat flux between the liquid and. vapor, and with various container wall properties. The outputs of major interest are the pressure rise, and temperature and velocity distributions. Representative plots of the isotherms and streamlines are included. The computations indicate that even though the container walls constitute an insignificant portion of the total heat capacity, less than 1% that of the liquid and vapor, variation of the wall heat capacity has a significant influence on the pressure rise rate. During a portion of the process, it is found that radial as well as axial stratification exists, with simultaneous evaporation and condensation occurring at various locations on the liquid-vapor interface. x

I. INTRODUCTION A number of space missions of current interest require the storage of liquid propellents for long periods of time, varying from hours to months. The ultimate goal is to maximize the quantity of useful propellent remaining at the end of this period of time. In the limit this maximum corresponds to the nonvented condition in which the original mass of propellent is retained. Whether nonventing is practical depends on the maximum internal pressure which results from the thermal interaction between the storage container, its contents, and its ambient. The prediction of pressure changes in a nonvented container is thus of major importance for purposes of feasibility determination and optimization. In Ref. 3 the results of the initial attempt to calculate the pressure rise in a single-component, two-phase system were presented for a two-dimensional case with axial symmetry and an imposed heat flux. The governing equations were placed in finite difference form and solved with an IBM-7090 computer, using the MAD language, for an example case with liquid oxygen. The measurements of pressure rise in a full size earth orbiting liquid hydrogen container on the Saturn IB vehicle AS-203, as reported in Ref. 17, makes possible a comparison with pressure rise calculated from this model. Certain deviations from the actual physical system are necessary in defining a model owing to the present stage of development of the computational procedures involved. These will be discussed at the appropriate places. This report presents the results of the computations made on a model simulating the S-IV B stage on the Saturn IB vehicle AS-203, along with comparisons with the measurements. Since the IBM-7090 computer had been replaced by the IBM-360 system at The University of Michigan, it was expedient to transform the source program from MAD to FORTRAN IV, and the computer programs are so presented herein. For the sake of completeness the general formulation presented in Ref. 3 is reproduced here. The pressure in a tank containing two phases, liquid and vapor, is related directly to the temperature at the liquid-vapor interface for a single component system. For a binary system, the pressure depends not only on the temperature at the interface but the relative liquid-vapor concentration of the two components as well. Pressure changes take place owing to heat and mass transfer interactions between the vapor, liquid, and container walls. The processes of heat and mass transfer interactions between the gas and liquid phases of a single component in cylinderical containers with axial symmetry are considered in this study. In the general formulation presented first attention is given to the cases of external pressurization with and with1

out liquid discharge as well as to the nonvented case. Solutions are then presented, utilizing numerical finite-difference procedures for the nonvented case. The initial conditions of the liquid and vapor are considered to be those at equilibrium, with uniform pressure P and saturation temperature T. From these initial conditions, the walls of the container undergo a thermal perturbation, such as a change in temperature or an exposure to an external heat flux, either of which may be an arbitrary function of time and axial location. The perturbations in the boundary conditions lead to a series of nonequilibrium phenomena within the container. Natural convection currents are set up in the liquid and in the vapor spaces. The liquid-vapor system tends to adjust to the new nonequilibrium conditions within the container by transferring mass and energy across the interface by either evaporation or condensation. The conditions at the liquid-vapor interface couple simultaneous transport processes in the liquid and gas phases. In the case of self-pressurization of nonvented tanks the rate of pressure rise within the tank is governed by the rate of heat and mass transfer from the liquid and the rate of heat transfer from the wall to the gas phase. The interfacial temperature is essentially that of equilibrium (saturation) conditions corresponding to the system pressure. At the same time, the temperature of the liquid-vapor interface affects both the interfacial mass and heat transfer, as well as the convective processes in both phases. These latter processes influence the temperature gradients within both phases and in turn will have an effect on the rate of pressure rise in the ullage space. Indeed, all the processes of heat transfer from the ambient to both phases, the natural convection within the container, the interfacial phenomena and the rate of ullage pressure rise are all mutually coupled. Such interactions have been the subject of many experimental investigations. Any analytical approach that would adequately describe the phenomena taking place in propellent tanks must take into account these interactions. This requires the calculation of the transient velocity and temperature profiles in both the vapor and liquid phases. Some work has been done along this line.l-3 The difficulties in solution of this problem is further complicated with the presence of turbulence or boiling of liquid near the tank walls, as their nature is not yet fully understood nor adequately described.5 Some assumptions are necessary at the present time for the construction of models which reasonably represent the practical situations. A survey which reviewed much of the available literature in this area, such as pressurization, stratification, and interfacial phenomena in fuel tanks, is given in Ref. 5. Several studies which are pertinent to the present one will be mentioned. Thomas and Morse6 and Knuth7 have considered the phase change of suddenly pressurized single-component, liquid-vapor system. Thomas and Morse6 presented both an exact solution and an approximate solution yielding an explicit expression for the mass transfer at the interface. Yang, et al.,8-10 have solved the problem of phase change in a suddenly pressurized single-component and binary liquid-vapor system. 2

Thermodynamic equilibrium is assumed at the interface, i.e., the interfacial temperature is always at the saturation temperature corresponding to the system pressure. Experience shows that this is a reasonable assumption.5 Epstein, et al., 11 used finite-differences for the calculation of the pressurization process. In this model, the axial variation of temperature is considered while the radial variation is neglected. The wall to fluid heat transfer was accounted for by introducing an effective heat transfer coefficient between the fluid and the wall. Also, effective thermal conductivities and mass diffusivities between the adjacent fluid layers were used to simulate the effects of random fluid motion. Provisions for variable tank cross-section as well as variable heat transfer coefficients were made in the program. The effect of boiling near the walls was not considered, and momentum effects were neglected. The principal advantage of this program is its simplicity and the relatively small amount of machine time required to advance the solution to a particular time level. However, the determination of the effective thermal conductivities and heat transfer coefficients and their variation with location require the determination of a large number of empirical constants, which require considerable experimental experience with a particular system. Although some analytical models have been used to study liquid stratification in self-pressurized propellant containers, with the exception of Ref.3, none has considered the simultaneous interactions between the liquid and the vapor phases. Most of the available studies of the pressure and temperature transients in the vapor space are experimental in nature and have served to identify some factors influencing the rate of pressure rise within the containers. The studies of Huntleyl2 show that liquid stratification causes the pressure to rise at a higher rate than that calculated using the average or mixed mean liquid temperature. These also showed that stirring the liquid causes a smaller rate of pressure rise, while stirring the vapor resulted in a substantial increase in pressure. Higher rates of pressure rise were obtained with smaller ullage volumes. Liebenberg and Edescuty have shown similar results.13 3

II. FORMULATION'A. GENERAL FORMULATION The formulation of the most general case will be given first. This involves external pressurization, liquid discharge, and the consideration of the vented and nonvented container. The heat capacity of the container side wall is also included in the formulation. The assumptions and limitations will be mentioned as appropriate and the results and problems encountered in the solution of the unvented, nondischarge case will be given. A cylinderical container of diameter 2a, height b and with wall thickness 56, is partially filled with a liquid as shown in Fig. 1. The initial height of the liquid is Xo, and that of the vapor is b-Xo. The origin of the coordinate system is taken at the center of the container base with x-positive in the direction of the liquid. At any time t, the location of the liquid-vapor interface is given by X = X(t). The fluid in both phases is initially at rest in an equilibrium state at a uniform temperature To and pressure Po. From these initial conditions the tank side walls are subjected to an arbitrary heat flux, qw(x,t). The tank ends are assumed adiabatic with zero heat capacity, but this restriction can be relaxed if desired. The differential equations describing the physical behavior are listed separately below for the various regions. 1. Tank Side Wall The differential equation describing the temperature-time history of the tank wall, considering the wall to be lumped radially but not axially, is,\ aT p -)w -(k + (k w + q(xt), (1) p C)tw C~f(")f W where the subscripts w and f refer to the wall and the fluid, respectively. The first term on the right-hand side of Eq. (1) is the rate of heat flow from the wall to the fluid, while the second term accounts for the net axial. conduction. Initial and Boundary Conditions T (x,o) = T (2) aT (ot) aT (b,t) ax ax

'xC b gIV~~~~~~~~~~~~~~~~~~~~~~r' a Fig. 1. Container configuration and coordinate system.

2. Liquid Region The following assumptions are made: (a) Thermal conductivity and viscosity are constant. (b) Fluid is incompressible. Density variations are introduced only in the body force term of the momentum equation. These variations are described by p = poll + (To - T)] (3) (c) The influence of viscous dissipation is neglected. Governing Equations (i) axial-momentum Du p u 1 u P Dt P-gg - r r (4) (ii) radial momentum Dv aP av v 1 TV 2vC (. (iii) continuity bu av v - + + - = 0 (6) (iv) energy DT T 1 aT Y2T\ Dt rx2 rr Or2/

Initial Conditions T(x,r,o) = T, u(x,r,o) = 0, v(x,r,o) = O (8) Boundary Conditions Velocity u(o,r,t) = O (9) w u(X, r t) 2 (10) where w d is the rate of liquid discharge. u(x,a,t) = O (11) u(x, o t) = O (12) 6r v(x,a,t) = 0 (13) v(x, o, t) = —0 (14) v(o,r,t) = 0 (15) av(X,r,t) _ O (16) Temperature T(X,r,t) = Ts(t) (17) T(x,a,t) = Tw(x,t) (18) w~~~~~~~~(S

aT( x, o, t) = (20) = o (20) br where Ts(t) is the saturation temperature corresponding to the system pressure and T is the wall temperature, Eq. (1). w 3. Vapor Region The differential equations for the vapor region are obtained assuming negligible changes in the viscosity and thermal conductivity within the vapor phase. The compressibility of the vapor is considered. Governing Equations (i) axial momentum Du aP Au 1 +au 8U\ 1 a +u av v2 p - = -pg - - + + +(21) Dt r orr r 5 Cxr r/ (ii) radial momentum Dv _P v V l v eVdbv\\ 1 + + (22) P Dt = 2 -r + rr r2 3 r r (rr (iii) continuity Dt P + + = (23) (iv) energy DT. D(P/p) DP (kT 1 aT T'\ pc -+p + k + —+ (24) v Dt Dt Dt r r r ) (24) (v) equation of state P = zpRT (25)

where z is the compressibility factor which can be evaluated from P, V, T data. Initial Conditions T(x,r,o) = T p(x,r,o) = po u(x,r,o) = 0 v(x,r,o) = 0 (26) Boundary Conditions Velocity u(b,r,t) = 0 (27) or w u(b,r,t) = u = -, (28) gp p ia gP where w is the rate of mass flow of the pressurant and p is its density. Equatio (28) is written for the case of uniform velocity at the inlet diffuser of the tank as in Fig. 2. u(X+,r,t) = ugi (29) u( x, o,t) = (30) =0u(xat)=O (31)

v(b,r,t) = 0 v(x,a,t) = 0 (33) v(x,o,t) =O (34) av(X,r,t) (35) where ugi is the velocity of the vapor at the interface caused by simultaneous phase change and liquid discharge, Eq. (45). The boundary conditions given by Eqs. (16) and (35) assume a zero shear stress in both the liquid and vapor at the interface. These are reasonable and are made for the sake of simplification. Another approximation would be to take the vapor velocity at the interface equal to that of the liquid at that point. The interfacial shear stress would then be determined by the analysis. For flows with large radial gas velocities at the interface, this latter boundary condition becomes more realistic. Temperature T(X,r,t) = T (t) (36) T(x, a., t) = T (x,t) (37) a\( x, o, t) t o= o (38) ar artb rnt) = O (39) or T(b,r,t) = Tgp(r,t) (40) Equation (39) applies for the case with no heat transfer across the top surface and no external pressurization, and Eq. (40) for arbitrary external pressurization. 10

wgp -Up -=+P A P CONTROL SURFACE VAPOR I I qj u q wg [W Wqi A I= dX -W d+Wi qw. L.QUI v! ui: dt P A W d Fig. 2. Control volume of vapor region.

4. Interfacial Heat and Mass Transfer The processes in the liquid and vapor regions are coupled by Ts in Eqs. (17) and (36) and by ugi in Eq. (29), which is related to the rate of mass transfer at the interface. The rate of mass transfer by evaporation or condensation across the liquid-vapor interface depends on the relative rates of heat transfer by diffusion from each phase at the interphase. Conservation of energy at the interface determines the rate of phase change as well as the interfacial velocity of displacement, according to; hfwi = a k T(Xrt) -[k ~T(X,r,t) iTrdr (41) fgi f([k, g where wi is the rate of interfacial phase change. According to Eq. (41), wi will be positive if condensation takes place. The interfacial displacement with no liquid discharge is then given by: dX pQea2 1t = Wi (42) P~a~ dt i Should it be desirable to include the process of liquid discharge, Eq. (42) would be written dX PQta2 dt =- (43) where wd is positive for liquid discharge, and dX dX dX i d u =-= + (44) i dt dt dt where dX. - = rate of interfacial displacement due to phase change, dt dX = rate of interfacial displacement due to liquid discharge, = combined rate of interfacial displacement due to phase change and liquid discharge. 12

Equation (44) is obtained assuming that the liquid surface remains flat during the discharge process. This is an approximation which neglects the influence of viscosity near the walls and surface tension effects. Such effects are negligible except for very low gravity levels and small Bond numbers. The vapor velocity ugi at the interface is related to the rate of interfacial displacement due to phase change and discharge by, - dX dX dX U =-2 + (45) gi ('g5) dt dt B. TRANSFORMATION OF THE PARTIAL DIFFERENTIAL EQUATIONS The momentum and the continuity equations are combined to obtain the vorticity equation and an elliptic equation relating the vorticity and the stream function. This is accomplished in the same manner as described in Refs. 1-3. The x-momentum is differentiated with respect to r, the r-momentum equation is differentiated with respect to x and the two combined to eliminate the pressure terms. 1. Liquid Region Equations (4) and (5) reduce to: Dct 1 +3 Dt pr x r r (46) where w' is given by: W' = 7 - (47) Introducing the stream function 4', defined by 1 54' 1 tI u -; v = - (48) r $r r ~x Equations (7), (46), and (6) are transformed respectively to: aT 1 Ir' aT 1' T +(T 1 T ) (49) at r ar ax r 2x ar x' 15

2 1 t +1 t = tr2 (51) O-lta r ~x arr r r The initial and boundary conditions must also be transformed appropriately. Equations (49), (50), and (51) are solved numerically by finite differences to obtain the temperature, vorticity, and stream function distributions. From Eq. (48) the velocity distributions can then be computed. 2. Vapor Region Combining the momentum Eqs. (21) and (22) in the manner described above, the following is obtained: Lu) ax] +Dp lvu l+ v~Pj g 2 + - (52) Dt p Dt pr Dt Or pr Dt x pr Or + r Cr r where a' is given by Eq. (47). Equation (52) is the vorticity equation for this case. The first terms within the brackets on the right-hand side of this equation account for the compressibility effects due to density variation. Should the compressibility effects be neglected, Eq. (52) will reduce to Eq. (46). It should be noted that the presence of these terms does not introduce any additional difficulties as far as the solution of the vorticity equation is concerned. The difficulty will be in combining the continuity Eq. (23) and Eq. (47), in order to obtain an equation similar to Eq. (51). However, this difficulty can be overcome if the term oap/t is neglected in combining the continuity Eq. (23) with the definition of vorticity. The rigorous justification for this approximation has not yet been established. In this case Eq. (23) will be rewritten as: d(pu) 1 a(prv) =. (53) ax r Or The continuity Eq. (53) is combined with Eq. (47) by introducing the stream function T", which satisfies (53). This stream function *" is defined by: 1 adt", 1 a r" u -; v: - (54) pr Or pr ax 1_4

Substituting (54) and (47) into (53), the following equation which relates *" to W', is obtained, a2 r pr + x -' + r Lu v P (55) The last terms within the brackets in Eq. (55) account for the density changes in the vapor region. The terms ap/~r and ap/~x are calculated in the finite-difference procedure using the equation of state and the temperature distribution from the previous time step. The coefficients u and v in u ap/lr and v aP,/&x will be taken equal to their values at the previous time step, which is the same procedure used in calculating the nonlinear terms u aT/x, v aT/&r, u ak'/?ax, and v c'/ar, thus in effect linearizing these terms. The spatial variation of pressure in the vapor region can be neglected. The density terms in Eqs. (52) and (55) are evaluated using the equation of state (25), and the derivatives given by: ap TP(I ZT5 aT (56) Cp P (ZT) aT R2 (57) Substitution in Eq. (52) gives: Dc) cW' dP +' a ZT) DT Z aT +1 + T ZT) Dt P dt ZT\ aT j Dt rZT\QT r rZT ar Dt rZTk \T x Dt L x r r J (58) Substitution of Eqs. (25), (56), and (57) into Eq. (55) gives: C,, " _2p P,r2 Pr _ZT ) ~ - r a kr x -2 RZ -T' R-]p r (59) The terms u and v in Eqs. (58) and (59) could also be expressed in terms of," by Eq. (54). Substitution of the equation of state (25) into the energy Eq. (24) gives: 15

r4ZT5\1 DT dP Ip[+(8:D= + kVRT (60) P Taking the thermodynamic relation C v =T(T() (61) P v and the ratio of specific heats C -C (62) v and substituting into Eq. (60) and rearranging, another form of the energy equation is: DT 1 RZT dP Y7 T (63) Dt 1 + (Y - 1)Z C P dt 1)Z Q(ZT) V 1+ (ZT) V It might be noted that the body force term in Eq. (58) could also be expressed as: 9g (ZT T (64) rZT\ jT / _ rT For the case where the behavior of the vapor is approximated adequately byanidealgas, Eqs. (58), (59), and (63) reduce, respectively, to: Dco' gc' dP cO' DT 6g T 1 dT Du 1 dT Dv C'co' l3' Dt P dt T Dt rT dr rT rDt rTx Dt x r ar arJ (65) -2t,, }",,, pt t r 2 Pr r f T ar2 r a RT RT ar V ax(6) 16

DT y - 1T dP+ T (6) UV' J2T (67) Dt y P dt Equations (58), (59) and (63) (or (65)-(67) for ideal gases) together with the transformed boundary conditions constitute a system of 3 equations with 4 unknowns for the vapor space, c'(x,r,t), *"(x,r,t), T(x,r,t), and P(t). The finite-difference solution of Eqs. (58), (59), and (63) for w', *", and T requires that a function for dP/dt be available. This is obtained from the First Law of Thermodynamics written as an instantaneous rate equation, taking the vapor space as the control volume. Referring to Fig. 2 this formulation is written: d dV dt g(t)(p e )dV - A. hgsgs (ugi - u.)dA + h p u dA = qdA - P d dt V (t) g g Ai gs gs gi i A gp gp gp Ac.s, dt 1 p c.s. (68) This formulation includes the following generalized variation: e = e(x,r,t) P = p(x,r,t) g ugi = u(r,t) h gs are uniform over the liquid-vapor interface gs gs u. is uniform for a flat interface h,p,u are specified by the pressurization process gp gp gp q is the local heat flux on the control surface, and its integral includes that from the tank wall and through the liquid-vapor interface. Given property information on eg = fl(P,T), pg = f2(PT), hgs = f3(P), P = 4(P), the desired term dP/dt can be extracted from the first term of Eq. (98). The necessary properties can come either from specific heat data plus an equation of state, or from tabulated values. It may be anticipated that different computational procedures would be required for each of these. 17

The procedure will be demonstrated for the case of self-pressurization, that is with no liquid discharge, no external pressurization, and a flat interface. Thus, wd = O, wp = 0, and up = O. Then, dX dX i =u (69) dt i dt and from Eq. (45), assuming a uniform vapor velocity across the interface u - u = - - - u. (70) gi i p - p gi gs For the coordinate frame selected dV _ dA - A - -Au (71) dt i dt i i For brevity qtot fA qdA (72) CoS. Substituting Eqs. (70)-(72) into Eq. (68) d P dt V() (P e )dV = qt + h p UgiA. + PA u. (73) dt V (t) g g tot gs gs p - p gii ii g gs Using Leibnitz's rule for differentiation of integrals having variable limits the first term of Eq. (73) can be written as de ap d (P e )dV = p dV + f e dV - p e u.A. (74) dt v g( g g V =g t gs gs 1 1 g Vg g Substituting Eq. (74) in Eq. (73) and rearranging with the use of Eq. (45) gives, -e ap. IV P - dV + f e - dV = q + h A u (p - p ) (75) g t g Vg t tot gsii( gs 2 g g 18

For the case in which the use of an equation of state (25) is desired, Eq. (75) can be expressed as the following by means of general thermodynamic property relations: v dP dV v _ V g) a__ZT) R' Z2T2 = qtot gs i i gs The enthalpy terms in Eqs. (75) and (76) can be expressed by: h-h =T C dT + f P R - [ ) AP (77) R f- R P R PR where the subscript R refers to a low pressure reference condition. For an ideal gas with constant specific heat Eq. (76) reduces to: v dP _(-)v =q + h AiU(p -p) (78) R dt g tot gsii gs Expressing the enthalpy in terms of specific heat and rearranging, Eq. (78) reduces to: dP R + RT 9^ A.u(p - P) (79) dt CV V qtot V i gs v g g For the case far from the region of the critical state, such that p >> Pgs' Eq. (79) can be written as dP R 1 8 s) dt CV tot V v g g where w. is the rate of interfacial mass transfer (having a negative value for evaporation) and is given by PP wi =- p A.(u. - u -) = -- Ags u gsi gi (pi -p ) ig (81) gs (P ~ Pgs 1 gi 19

C. DIMENSIONLESS FORM OF THE EQUATIONS The governing equations for the wall, liquid, and vapor regions (1), (49), (50), (51), (58), (59), and (63), along with the boundary conditions, are restated in dimensionless forms for convenience. The substitution used to nondimensionalize these are: ab ava u - a2 U; v - V; T-T - ~a a o ga t = -; x = b; r = ar etb " = Qh c,; r' = 4 ab~r (82) a The resulting dimensionless equations are given below. 1. Tanic Side Wall The dimensionless equations describing the wall temperature, from Eq. (1), are given by: (i) O<_ < i b (pC ) OQ a2 PC k dPG w a p w ~-(ii) b+ c > 1 w (pCp) k(() +2 b (PCp) kw i + a. -. Prw Grw(b t) (83) 0'r a(pC) Pa2 (pC) k w2 a a (PC,)e * w p w 20

where Gr (t,t) is the modified Grashof number and is given by: w Gr*(r,t) = ka qwt) (85) (iii) Boundary and initial conditions`\ (O,-) a6 w ( 0(1,T) = ~ (86) e(J,o) = o (87) 2. Liquid Region (i) The energy equation, Eq. (49) a i ab a 1 ab c' a2 20e 1 a a2Se + ~_ + —- (88) a a a b' rl a (ii) The vorticity equation, Eq. (50) aS+ 1 av aS 1 at aS 1 ao a2 a2Wc 3 at, a2( =Pr a+ + 3+ (89) at TJarl an: r, a: arl Q~rl a 7 b at: Tl (iii) The vorticity-stream function equation, Eq. (51) T]2 (S l a+ a2t = (90) u; v = (91) (iv) The boundary conditions (1) Stream function boundary conditions o e*(o,,I)IT (OlT)I = = 0 (92) ~F ( C, T, T) ( 1, ) = a = o (93) 21

r (s l.T) = a- (ta,')]= (94) 1(, 1i ) =ai,, ) = 0 (95) (2) Vorticity boundary conditions dn(l,), T) = 0 (96) d( A,OT) = 0 (97) Two additional vorticity boundary conditions are required at the tank wall and bottom. An explicit expression for the vorticity at any of these locations is difficult to obtain. The method of solution used in this work, which follows that of Refs. 1-3, overcomes this difficulty. (3) Thermal boundary conditions G0(,1,-) = G (',-) (98) G( i,,-T) = G ()) (99) t.0l, ) = 0 (o100) (M(r,0,') O (101) acn where 0 is given by: s _ a_ (T) = Pr IT (t) - T2 w (102) s sb (4) Initial conditions ( 5,r,o0) = u(,,o ) = v(,r,o) = c, r,,O) = 4( ~, r,0) = o (10o3) 22

3. Vapor Region (i) The energy equation, Eq. (63) "le 1 a a 1 a ao 1 RZ (+0) dP aT P., a,, aC prl am n 1 + - l)Z/((Z)) v Y - V)a aa 3T v 1 + ( - 1)z/( A(ZT)) PLb 2' + 2 a,2 v (104) where 4 g a q3= T (105) ~ VI0 Cb o (ii) The vorticity equation, Eq. (58) aCi 1 a_ __ 1___ a_ g a _ -- +...= Pr 5+-] — 4P- + +) + e1 + hsZT be rZ(DU 1 ZTT a DV+ (dPT TjZ( + G )\ 3T JP aTZ DT + G kTJ D P dT (106) where p has been replaced by the term: 1 VZT)) (107) as in Eq. (64). (iii) The vorticity-stream function equation, Eq. (59) 23

b a2 a ZR( + 0) + - Ba_ b2 a52 T ZR(G +e 2 g+T-a L[ UAnb PV (108) v a b Z2R(G + 111U b - (iv) Boundary conditions (1) Stream-function boundary conditions -= am( I'1) = 0 (109) p U 2 w 2,2 gs pa 2Jtb 4(M,(O,) 1K X( T) O (111),( ~,CoT) = T) 0 (112) (2) Vorticity boundary conditions CW(,O,T) = 0 (113a) W( ij~,?) = O (113b) (3) Temperature boundary conditions )( i,T,) = "G (T) (114) a = o (115) 24

9(t,1,) = G (,uT) (116) (1,0,) = O,0 (117) (4) Initial conditions Go(I,r,o ) = U(C,r),O) = V(!,r,O) = r(,,o) = c(,,o ) = o (118) From the above results, it is clear that the temperature, pressure, and velocity within the container are functions of the parameters Gr*, Pr, (pC ) /(pCP) (k /k,) (ag/a) (Vg/Ve), ( g/%), (a/b), and (a/8). Gr* is defined inP Eq. (85). g 25

III. METHOD OF SOLUTION A. FINITE-DIFFERENCE FORMS The finite-difference method of solution used in Refs. 1-2 is adopted here. A complete discussion of the application of difference methods for the solution of the energy and vorticity equations is given inthese references, along with the problem of stability of the difference equations. A brief discussion of the method of solution will be made here. The basic technique in the application of finite-difference methods for the solution of partial differential equations is the use of Taylor Series Expansion to approximate the derivatives at a point in terms of the value of the function at that point and/or at its neighboring points. This may be demonstrated as follows: 1. The time derivative is represented by af f(T + AT) - f(T) + o(A~) (119) aT AT 2. The first order derivatives af/ao and af/l can be approximated by (i) AftT) (,) + O(A~) (forward differences) (120) or ( f fa(,j,),) -,f,T) + O(A ) (backward differences) (121) or f f r + A,rT), - f( A - A{,n, T) + o(A )2 (central differences) (iii) ~~~~~ ~~2A~~ ~(122) 3. The second order derivatives are replaced by finite-differences according to the formula. a f fr( + A,rj,T) - 2f(T,r,,) + f(~ - A,,T) + o()2 (125) (__ -.(.52 26

The function f represents either 9 or w, and A1 is the spatial increment in the (-direction. The last term on the righthand side of Eqs. (119) through (123) indicates the order of the truncation error involved in replacing the derivatives by finite-differences. It is clear that the central differences offer a better representation of the first order derivative af/l than the forward or backward differences. However, the form used to approximate the first order derivatives is usually determined by stability considerations. Similar formulations can be written for the derivatives af/N and Yf/ln2. The substitution of the above formulae in the energy and vorticity equations produces a set of explicit difference equations. However, if the values of the function f in Eqs. (120) through (123) are taken at the time level T + AT instead of being taken at time level T, the resulting finite-difference methods may require the use of small time increments and consequently large machine time. Certain implicit formulations may permit the use of large increments. The application of both explicit and implicit methods to the present problem has been extensively investigated in Ref. 2. It was concluded that the lack of explicit boundary conditions for the vorticity at the solid boundaries prevent the use of large time increments, i.e., implicit methods. Other than that, it was decided to employ explicit methods. It is clear from Eqs. (120) through (122) that more than one explicit finite-difference formulation can be constructed for each of the vorticity and energy equations. The finite-difference formulation chosen for the solution of the present problem is dictated by stability, as well as practical considerations, which will be shown below in studying the stability of the finitedifference equations. The method of solution used in the present problem can be summarized as follows: 1. The time derivatives aG/a~ and ck0/aT are approximated by Eq. (119). 2. The nonlinear terms U 3/~, V 6/8, U 8X/8(, and V dao/~ are linearized by considering the velocity components U and V to be known and are taken equal to their values at time level T. The order of the error introduced by this linearization can be obtained from Taylor Series Expansion. If Uo and U are the values of the axial velocity component at time levels T0 and T0 + AT, respectively, then U a= U + A+ O(A ) + O(AT) (124) The second term of the right side of Eq. (124) represents the linearization error. 27

3. The nonlinear terms U ~/~ and U daw/2 are approximated by backward differences, Eq. (121), if the coefficient velocity U is positive and by forward differences, Eq. (120), if U is negative. The same procedure is followed for approximating the terms V G/ M and V dw/n according to the sign of the velocity component V. 4. Central differences are used to approximate the first order terms 1/r I9/3, 3/ /w/&q, and 1/rj 4r/8P. No stability problems will be encountered in this case. At the centerline where both rq and 6O/3 are zero, 1/q oQ/3q is replaced by its limit according to L'Hospital's rule Limit - a (125) 11 T7 TI~T2 =0 5. The second order term 20/a22 2e/a2 o~/a2 2 /2, /a 2 2 and 8 t/ 2 are represented by Eq. (123). 6. Although the first order derivative aQ/8 |r=l in Eqs. (83) and (84) can be represented by any of Eqs. (120), (121), or (122), the following formula, which has a higher order truncation error, is used, 0al, 1(r,1) - 180(r,j - An) + 9G(,, - 2Aq) - 2G(,q - 3AP) a r=l 6AnT (126) Similarly Ca ile(~,~) - 18e((. - A,r ) + 90(ti - 2A,r) - 20(~i - 3a,~) a: ~=Si 6nA (127) 28

In determining the side wall temperature at the location of the interface by the finite-difference procedure, a special difference equation must be used in order to take proper physical account of the possible axial variation of heat flux imposed on the wall, the difference in thermal conductivity between the liquid and vapor, and the possible difference in axial grid spacing in the liquid and vapor regions. The energy equation for the nodal point in the wall at the liquid-vapor interface is: wi (T~AT) - wi(T) kA + k A: ~wi wi (~ ~ (= ~~I g g AT C 6c~ J AT 2 2 pw pw 2 ll@(~i, ) - 18@(i,ryw-An) + 90(KyTw-2Al) - 2G(~i w-w5AA) 6AB 2 aw e(S,-enS~)+ ) +A g + (4 \(A+A) ) (i' -) -+ (i) +- (i y) b g 2 6g a6 + (qA~ qX A)g v bC w (128) 7. As mentioned earlier, two vorticity boundary conditions each are required in the liquid and vapor regions in addition to those given by Eqs. (96) and (97) for the liquid and Eqs. (113a) and (113b) for the vapor. An explicit expression is difficult to find, but the procedure followed in the numerical computations is described here. The step-by-step explicit computations of vorticity allow progressing from one vorticity distribution to the next at all grid points except those on the boundary, using the values of the vorticity from the previous time step. The new values of the vorticity are used to determine the stream function, which in turn are used to compute the vorticity at the solid boundaries. Using the Taylor series expansion together with the stream function boundary conditions, the following expressions are obtained for the vorticity at the solid boundaries: Sidewall 8G( ~,r %-Arq) - *(~,'%-2Aq) )=( 2 (9) 29

Bottom (or top) (o,r) -= ia2 8*(o+At,r) - *(o+2A,i) (150) - 2b2 $(a~>r B. STABILITY CRITERIA The stability of the finite-difference equations is an important consideration in establishing the size of the grid and the time steps, and the form of difference. The details are given in Ref. 2. As might be anticipated, the size of the grid and time steps will also be governed by the storage capacity of the machine and limitations of time and cost. The necessary requirements for stability are given by the following: /U. I Iv. 2a2 + 2 + i +- ) < 1 (11) AT(2 + (n)2 + n + <n (132) For the centerline, where V = 0,. AT a 2 <1 (133) 2a2Pr 4 Pr J1i\ I AT 2+ + 22 )2 < 1 (134) whichever one of the above is the most restrictive must be used in calculating the maximum value of AT permitted. C. COMPUTATIONAL PROCEDURES The sequence of steps in establishing the numerical calculation is as follows: (1) A convenient grid size is selected, limited by the machine storage capacity. (2) A suitable time increment is chosen. This may be adjusted during the process of computation as necessary to maintain numerical stability. 50

(3) The temperature distribution is computed using velocities, temperatures, and pressure from the previous time step. (4) These temperatures are used to compute the vorticity at the interior nodal points at the current time step. (5) The stream function is computed at the interior nodal points. (6) The vorticities at the solid boundaries are calculated using computed stream functions. (7) The velocity components are calculated. (8) The rate of phase change at the interface is determined using the computed temperature distribution, with Eq. (41). (9) The pressure rise is computed with an equation such as (80) using the parameters from the previous time step. (10) The above procedures are repeated successively. A problem inherent in the use of numerical methods is the accurate determination of temperature gradients. In the present application temperature gradients are computed in the liquid and vapor at the liquid-vapor interface to determine the rate of phase change, from Eq. (41), and in the liquid and vapor at the wall to determine the heat transfer to the bulk liquid and vapor, respectively. The general problem of the effect of grid size on the interfacial heat and mass transfer is presented in some detail here. The formulation with Eq. (41) was used by a number of investigators6-11 to determine the rate of interracial phase change for the case of a suddenly pressurized one-dimensional model. However, the determination of the temperature gradients by numerical representation of the calculated temperature distribution in the case of self-pressurized containers may be difficult for twodimensional cases because of the increased machine storage requirements combined with large temperature gradients near the interface, which cause the phase changes. These large temperature gradients exist in a very thin layer near the interface, as has been shown by experimental measurements.l4,15 This requires the use of a very small grid size in order to obtain an acceptable approximation for the temperature gradients near the interface. 31

In the case of the temperature gradients in the fluid adjacent to the wall two aspects must be considered when either the heat flux or the wall temperatures are imposed. One is similar to the liquid-vapor interface problem in that the grid spacing should again be small enough to permit sufficiently accurate representation of the temperature distribution in the fluid, either liquid or vapor, and in general dictates that the grid spacing be as small as possible. The degree of being sufficiently small will depend upon the magnitude of the imposed heat flux or the imposed temperature, and the response of the transient convective process to these disturbances under the prevailing effective gravity level. The only true test available is to vary the grid size for given conditions and compare the coupled results. The other aspect to be considered is a physical one involving only the liquid, but is also related to the problem of grid spacing. If the temperature of the solid wall in contact with the liquid exceeds the saturation temperature by some amount, dependent upon various parameters including liquid and solid properties and the configuration, nucleate boiling will be initiated. This particular heating surface superheat might be called the incipient boiling point. If information on the incipient boiling point is available for the prevailing conditions, an imposed wall temperature below this point then represents no additional problem beyond that of having sufficiently small grid sizes, as discussed above. For the case of a heat flux imposed on the outer surface of the container, however, the resulting wall temperature is a variable dependent upon a number of parameters such as wall thickness and heat capacity, fluid properties, acceleration level, and container geometry. Whether the wall temperature will exceed the incipient boiling point will not be known a priori, since the wall temperature is computed during the course of the computations. The procedure by which the possibility of nucleate boiling is taken into account is based on the following physical assumptions: (1) should nucleate boiling begin, further increases in heat flux generally result in relatively small increases in surface temperature as compared to nonboiling convection. This has been observed widely (e.g., Ref. 16). (2) The vapor bubbles formed are transported by buoyant forces to the ullage volume quite rapidly. The extent to which this may occur is as yet uncertain, but may be anticipated to depend upon the degree of subcooling present, the pressure and the effective gravity level. These are implemented in the computational procedure by considering that should the tank wall temperature, and hence the liquid adjacent to the wall, exceed the existing saturation temperature by some arbitrary amount, this excess is eliminated by the evaporation of the appropriate amount of liquid directly into the ullage space. In effect, then, a portion of the vapor bypasses the liquid-vapor interface. The arbitrary amount referred to above, the symbol for which is given as ATwmax might be considered as the incipient boiling point, and no longer will be arbitrary when sufficient information on its behavior is available. 32

The physical phenomena described above is simulated in the computer program as follows: (1) The container wall temperature is calculated using Eq. (1). (2) The liquid temperature is obtained using Eq. (49). (3) The calculated wall temperature in the liquid region is examined. If it exceeds the saturation temperature by more than the prescribed temperaturedifference ATwmax, it then is reduced such that it equals the saturation temperature plus the prescribed temperature difference. (4) Part of the heat added to the liquid region appears as enthalpy in the wall and liquid and the rest is used for evaporating some of the liquid. The portion of the heat transferred to the liquid and resulting in evaporation is determined by setting an energy balance according to x a aX Jo 2 aqw(x,t) dxdt + Jo 2Trq idr = Jo Jo 2pcp rr - dxdr aT X w + 2TraFpc bdx +wh O 2apw pwat ifg (135) where q is the rate of heat flow from the interface to the liquid and is given by, q = k I =X (136) Equations (135) and (136) are used to determine the rate of evaporation from the interface, w,. If the difference between the wall temperature and the saturation temperature is less than the specified maximum, ATwmax, then the procedure above is bypassed and computations proceed as described earlier. An implicit assumption in the use of this procedure is that the lamindr flow conditions described by the momentum and energy equations are not affected. This may be reasonable if the container is relatively large compared to the "bubble boundary layer" region next to the wall. In other words, if the vapor bubbles remain in the vicinity of the wall, the major bulk laminar motion of the liquid will not be influenced by their movement to the ullage space. An accurate physical description of this behavior requires additional analytical and experimental investigation in incipient boiling and the departure and motion of vapor bubbles 35

under low gravity fields with various patterns of subcooling. D.- COMPUTER PROGRAM The simplified computer program flow chart based on the formulation presented is given in Appendix A. Appendix B shows the computer program statement list for the case with liquid hydrogen and with the assumption of ideal gas behavior for the vapor. FORTRAN IV language is used here. In Appendix C are listed the meanings and units of the computer symbols employed. Appendix D lists the inputs (variables) required for the program, along with the units, and Appendix E presents a typical output. The compiled results for a number of computer runs covering several variables are presented and discussed in Section IV for liquid hydrogen. 34

IV. RESULTS Computations were carried out with the program listing of Appendix B, using values of inputs calculated from the telemetered flight data of the Saturn IB vehicle AS-203, as reported in Ref. 17, so as to compare the calculated pressure rise with the measurements. Not only do differences exist between the actual physical system and the model which serves as the basis for the computer solution, but uncertainties exist in the accuracy of the necessary computer inputs given in Ref. 17, thus making a realistic comparison between the measured and predicted pressure rises difficult at this time. In order to indicate the possible influence of the more significant uncertain inputs, these-are treated as variables in several cases. A. GENERAL ASSUMPTIONS The general assumptions incorporated into the particular program of Appendix B are listed below: 1. The tank is cylindrical with flat ends. 2. Acceleration or body forces act along the axis. 3. Two-dimensional conditions prevail, with variations only along the axis and radially. 4. Flow conditions are laminar within the entire container. 5. The vapor behaves as an ideal gas. 6. The liquid has constant properties except in the body force terms. 7. The tank side wall is uniform in thickness and has constant properties. The wall is lumped in the radial direction but axial conduction is taken into consideration. 8. The imposed heat flux on the outside of the tank wall is uniform, but differs in those portions in contact with the liquid and vapor. 9. The ends of the tank are adiabatic, and the heat capacity of the ends is neglected. 10. The grid size varies in the liquid and vapor region as liquid fraction changes, in order that a nodal point always exist at the liquid-vapor interface. 35

11. Initial conditions of uniform temperature and zero velocity exists within the container. 12. The Bond number is sufficiently large that a. reasonable approximation to a flat liquid-vapor interface exists. B. VARIATIONS POSSIBLE IN PROGRAM By relatively minor modifications to the program, some of the above listed general assumptions can be relaxed, providing additional flexibility. In any case, however, axial symmetry must be maintained in order that the problem be two- dimensional. 1. The imposed heat flux can be varied axially and with time. 2. Specified initial conditions of temperature and velocity can be utilized. 3. Specified heat flux to the tank ends, including variations with radius and time, and including heat capacity can be incorporated. 4. If the description of the physical process warrants, imposed temperatures of the container walls can be utilized, with variation axially and with time. 5. Axial variation of tank side wall thickness and variation of specific heat with temperature can be accounted for. 6. Radial variations in the tank wall temperature can be taken into consideration. This may be particularly desirable if the wall is of composite construction. Although the influence of variations in liquid properties with temperature and pressure can be incorporated with minor changes, the use of real gas properties in the vapor space will require major modifications to the program. This would be necessary should the pressure variation be large or should the pressure approach the critical state. Major modifications also are necessary to handle the spacewise variation of grid size in either the liquid or vapor domains. C. INPUT PARAMETERS Appendix D lists the input requirements for the program as presented in Appendix B. For purposes of presentation and discussion the inputs are subdivided by categories of geometry, heat flux, acceleration level (for body

forces), fluid properties, and miscellaneous. 1. Geometry The S-IVB LH2 tank is cylindrical with hemispherical ends outward at the upper end and inward at the lower end. The computer model here consists of a cylindrical tank with flat ends. For purposes of simulation it was decided to maintain the diameter, total volume, and ullage volume approximately the same between the physical system and the model. This makes the wetted area of the model considerably less than in the physical system, and cognizance of this should be taken in interpreting the results. The significant geometrical comparisons are shown in Table I. TABLE I COMPARISON OF GEOMETRY FOR LH2 TANKS S-IVB Computer Model Diameter 21.6 ft 22.0 ft Total volume 10,500 ft3 11,050 ft3 Ullage fraction 0.66 0.677 LH2 mass (initial) 16,000 Ibm 15,700 ibm Gas H2 mass (initial) 500 ibm 500 lbm Liquid volume 3,570 ft3 3,570 ft3 Height of tank 36.4 ft 29.2 ft Wetted side wall area 952 ft2 651 ft2 Wetted bottom area 505 ft2 379 ft2 Total wetted area 1,457 ft2 1,030 ft2 Unwetted side wall area 1,528 ft2 1,365 ft2 Unwetted top area 430 ft2 379 ft2 Total unwetted area 1,958 ft2 1,744 ft2 Total skin area 3,415 ft2 2,774 ft2 The tank side wall of the S-IVB consists (Ref. 22) of a lamination of polyurethane foam (with glass fibers and glass cloth on inside) and 0.134 in. thick aluminum on the outside, with reinforcing ribs 0.10 in. thick by 0.650 in. high on a 9.5 in.2 waffle pattern. Since the model as presently constituted considers only a radially lumped (thermally) wall, the composite properties of the physical system cannot properly be taken into account, and requires a modification of the program. For purposes of this study, a radially lumped wall was assumed, using in one case the properties of the insulating layer of polyurethane foam alone, and in the other case the uninsulated aluminum skin alone, giving due regard for the axial conductance path contribution of the 37

reinforcing ribs. The effect of each of these models is to make the wall temperature more nonuniform in the axial direction than would be the case with a composite wall model, and results in a different distribution of energy input rate between the liquid and vapor. The relevant properties used are listed in Table II. As a further test of the influence of wall material, one computation was conducted for stainless steel with the same thickness as the aluminum. TABLE II TANK WALL PROPERTIES Material Polyurethane Aluminum Stainless Steel Thickness, b 0.71 in. 0.140 in. 0.140 in. Density, Pw 10 lbm/ft3 169 lbm/ft3 488 lbm/ft3 Specific heat, Cpw 0.35 Btu/lbm-~R 0.20 Btu/lbm-~R 0.11 Btu/lbm-~R Thermal conductivity, kw 0.025 Btu/hr-ft-~R- 219.5 Btu/hr-ft-~R 8.0 Btu/hr-ft-~R Thermal diffusivity, aw 0.1985x10'5 ft2/sec 1.675x10o2 ft2/sec 4.16xlO'5 ft2/sec Total heat capacity, pCpb 0.207 Btu/~F-ft2 0.394 Btu/~F-ft2 0.626 Btu/OF-ft2 The properties listed above apply for near room temperature, which is valid if a polyurethane layer of insulation lies between the cryogenic liquid and the outer metal skin. A number of computer runs were also conducted with the aluminum wall case, but using a heat capacity equal to 1/10 of that tabulated above. This in effect makes the heat transfer rate to the tank contents more nearly equal to that on the exterior surface of the tank, but retains the axial conduction effects of the tank side walls. The desirability of minimizing the tank wall heat capacity arises because the heating rate is given to the tank contents, excluding the tank walls. 2. Heat Flux In Section VII of Ref. 17, the mean rates of energy input to the liquid and gaseous contents of the fuel tank are computed from the physical measurements by four different procedures. These heating rates are summarized in Table III and the computation procedures discussed are below. The values by procedure A are based on an energy balance using the temperature measurements in the liquid and vapor during the closed tank experiment. In this procedure it was assumed that all of the energy required for evaporation during this period comes from the liquid alone. 38

TABLE III S-IVB FUEL TANK HEATING RATES Liquid, Ullage, Total, Procedure Btu/sec Btu/sec Btu/sec A 29.9 5.9 35.8 B 23.5 11.5 35.0 C 21.0 6.6 27.6 D 13.5 17.0 30.5 Procedure B is based on the continuous vent data, using the mean values from Figs. VII-2 and VII-3 in Ref. 17. It might be anticipated that some difference in the pressure rise would arise between using a mean value and a periodic heating rate as in Fig. VII-2 owing to the nonlinearity of the system. Procedure C is based on a combination of the continuous vent data and the fluid temperatures measured during a. transient period between the first and second blowdown. The heating rates with procedure D are computed by using the temperature measurements across the tank walls to calculate the wall heat flux, using previously determined values of the thermal conductivity of the wall material. It is noted that these different procedures give results ranging from approximately equal energy input rates to the liquid and vapor, to liquid rate five times that to the vapor. If the primary source of energy is solar radiation, the heat flux should be distributed almost uniformly on the tank exterior, in the axial direction. The higher heating rate to the liquid might then be considered as due to axial conduction from the vapor region to the liquid region, as a result of the lower heat transfer rate between the inner tank wall -and the vapor compared to that between the tank wall and the liquid. In the computer model the total energy input rate of 35.8 Btu/sec, corresponding to procedure A in Table III was used. In one case this is distributed between the liquid and vapor as in procedure A, and in the other case it is distributed such that the heat flux is uniform over the entire exterior of the container wall. Using the side wall areas of Table I, these result in the distributions as shown in Table IV. The heat fluxes listed are exterior to the surface, and owing to the heat capacity of the tank walls it can be expected that the energy input rate to the contents of the tank will be much lower than that of the actual case which this model simulates, with a lower pressure rise. 39

TABLE IV COMPUTER MODEL HEAT FLUX DISTRIBUTION Heating Rate, Btu/sec Heat Flux, Btu/hr-ft2 Case Liquid Ullage Total Liquid Ullage A 29.9 5.9 35.8 165.5 15.6 E 11.6 24.3 35.8 64.0 64.0 3. Acceleration From Fig. IV-13 of Ref. 17 it is noted that during the closed tank experiment, the vehicle acceleration decreased approximately exponentially from a/go = 3-7 x 10-4 at the beginning to a/go = 0.8 x 10-4 at the end. For determining the influence of other parameters an approximate average value of a/go = 1.7 x 10-4 was used. However, to provide an indication of the significance of the body forces, computations were made with values of a/go above and below this level at 2.5 x 10-4 and 1.0 x 10-4. 4. Fluid Properties The properties of hydrogen used in the computer model were obtained in Refs. 17-19. Those listed here are taken as constant, corresponding to the saturation temperature at the initial pressure of 12.4 lbf/in.2. Liquid Hydrogen: Thermal conductivity, k = 1.89 x 10-5 Btu/sec-ft-OR Coefficient of expansion, Pi = 0.862 x 10-2 ~Rl Kinematic viscosity, v = 2.065 x 10-6 ft2/sec Specific heat, C = 2.6 Btu/lbm-~R Thermal diffusivity, ao = 16.6 x 10-7 ft2/sec Density, pn = 4.39 lbm/ft3 40

Gaseous Hydrogen: -6 Thermal conductivity, k = 2.5 x 10 Btu/sec-ft-~R v Kinematic viscosity, v = 0.85 x 10 ft2/sec v Specific heat, C = 2.85 Btu/lbm-~R pv Specific heat, Cv = 1.492 Btu/lbm-~R -5 Thermal diffusivity, a = 1.06 x 10 ft2/sec v Gas constant, R = 766.4 ft-lbf/lbm-~R To evaluate the saturation temperature corresponding to the system pressure the following correlation and constants (Ref. 20) are used. loglo P = A + + DT (137) C +T where P = vapor pressure in atm T = temperature in ~K A = 2.000620 B = -50.09708 C = 1.0044 -2 D = 1.74849 x 10 Data on the latent heat of vaporization in the range of interest are read in tabular form.21 5. Miscellaneous Within the limitations on the total number of nodal points, dictated by the machine storage capacity and other economic aspects of the computational process, one can adjust the distribution of nodal points between the liquid and vapor region somewhat arbitrarily. For most effective use, including an optimum accuracy, the nodal points should be more concentrated in regions of greater velocity and/or temperature gradients. In the present case, with 31 axial nodes present, and on ullage fraction of 2/3, the axial grid spacing was selected which gave a spacing in the liquid region 1/4 that in the vapor region, since it was felt that the largest changes would take place in the liquid region. In one case a run was repeated with equal axial grid spacing to test the influence of this parameter alone. The resulting differences were negligibly small. 41

An additional parameter, which must be specified is the value of ATwmaxv the arbitrary amount by which the side wall temperature cannot exceed the saturation temperature. This can be considered as the incipient boiling point. Coeling4 studied incipient pool boiling for cryogenic fluids, and reported that the initial vapor formation in liquid hydrogen and liquid nitrogen is primarily a function of the surface superheat, for a given surface —fluid combination, and is not a strong function of orientation or heat flux. The results of the measurements for 14 surface-orientation combinations were plotted and it was found that the heat flux and surface superheat of each individual observation was generally located along the natural convection correlations, Nu = 0.14 (Gr Pr)1/3, for all horizontal surface-liquid combinations. From this correlation, surface superheat is calculated as 2.3~R at 166 Btu/hr-ft2 for LH2. Due to the uncertainty, ATw max was assigned 1~R for the present work. On a surface consisting of a glass fiber material coated with an epoxy cement, similar to the inner lining of the LH2 tank on the S-IVB stage, it was reported4 that the surface became active at approximately 0.5~R superheat. D. RESULTS OF COMPUTATIONS In Table V are listed the various computational runs made, with a listing of the different independent variables used, and the significant dependent variables, or outputs, at different values of real time. These latter are the system pressure and the total mass of liquid evaporated. Reference to these runs in the following comparisons will be made in terms of the B-H numbers listed. Most of the column headings in Table V are self-explanatory. The Max. Real Time gives the maximum time for which a run was made, but is not the computer time. The Max. Time Steps gives the maximum number of time steps used in the numerical marching procedure, and gives an indication of the computer time used. 1, Typical Plots of Streamlines and Isotherms In Figs. 3a, b, and c are shown computer output plots of the non-dimensional isotherms in the liquid and vapor regions at three different times during the transient heating process in Run B-H 47. These dimensionless temperatures can be converted to real temperatures by means of Eq. (82). Although the computer grid contained 21 x 31 nodal points, for these graphical presentations these were increased by a factor of three in each direction by linear interpolation of the results such as shown in Appendix E. Each digit in the body of the plot corresponds to a range of temperatures within which the temperature at that point lies. The key for the ranges is given in the table at the bottom of each figure. Except for the digits 1 and 9, 42

TABLE V INDEX OF COMPUTER RUNS Total No. of Max. Max., wall Heat max. NO. max. 100 Run B-H Vertical a/9 (Q/A),r (q/A)L Wall CW Capacityi Real of Time Press., Total Total No. NO* Divisions ft2 Material Tiie, 2 Evap. i Press., Btu/br- Btu *F od Btu/b-ft-*F ft-2/ses steps Tbf/in. Evap..p Press A-m$a Liquid Btu/4-ft,2 see lbift lbfZin.2 1bm Xbf/ii I-1 4T 20 I-Vio) 15-55 165.5 Poly .35.207.025.1985 (lo)-3 4T3.2.2 3.644 18-5T 3.o6. 9 --- 12.T0 5 I-2 53 10 LT(10) 15-55 165-5 Poly. -35.20T W.1985(lo), 50W.4 u83:L9.5 123. --- --- 12.8i I-3 59 20 3-VIO) -4 15-55 10.5 Al 02 -0394 219.,5 oi8 5361.5 2586 qia6 62.7:5 12.5-4 I-4 63 20 I-VIO) 3.5-55 165.5 -Al.002.00394 219.5.18 6To.i n56 3.3-44 -0553: 12.45.00131 13.2! V-1 52 20 1-T(10-4 64. 64. P*3,T..35.20T.025.3-985 (lo) -5 5003. 3.093 3.6.5.6... -- 12.4(, -4 V-2 55 20 1-T(10) 64. 6.4. S*S. Ill.626 18.0 Ai6 (lo) 5360-T T77 14.8 19.3 3.2.6 -4 V-3 56 20 1.T(3-0) Al 00215.0425 219.5 ol675 2000.1 1023 15.5 6.8 13-01 -4 8.45 - 12.5i V-4 5T 20 1.7(lo) 64. 64. Al 2 -394 219,5.0018 5360. 955 15.1 -T-5 62 20 1.7(10) -4 64. 64. Al.002 -00394 219.5.18 I Ob. 853. 25.28 263-1 25.28 263.i -4 v-6* 71 20 1.7(10) 64. 64. Al 02.0394'219.5 oi8 2000.5 1551 l8-4T 64.3 3.2.42.03-13 3.2.91 ii-I 64 20 (lo)-4- 15-55 165-5 Al.02.0394 219.5 xlB 2501.T 814'15-65 32.7 3.2.41 -13T i2.8( IT-2 65 O` 2.5(10)- 3.5-55 165-5 A-l.02 i,0394 219-.5 oi8 3500-1 1937 IT-37 43.6 1,Al.13T 12.50 ixi-3,4* 66 20 1.7(10) 64. Al.0215.0425 219.5 m675 34021. 188i 20.52 1.468 12.8,1 *No.liquid va.11 emperature suppressing'. **This run only bas unage fraction of -333, but all others'have.667.

each digit, and the blank spaces between, represent equal ranges of temperature. To provide the maximum information it was necessary to change ranges between Figs. 3a and b. Thus, the digit 7 in Fig. 3a corresponds to the same range of temperatures as the digit 6 in Figs. 3b and c. In the liquid regions of Figs. 3a and b one can note the motion of the heated liquid adjacent to the wall as it rises toward the interface, and then turns inward, and then downward. As a result "stratification" can exist not only in the axial direction but radially as well, owing to the very low level of axial body force used. In the vapor region one notes the formation of one and then two vortices. From Fig. 3c, it is possible to have quite large temperature differences in the vapor at a given axial level, depending on the radius. These figures indicate that the liquid is superheated to some extent almost throughout the entire domain, and axial stratification is almost nonexistent for these conditions. The pressure rise rate, which is related to the temperature of the liquid-vapor interface, is thus much lower than if axial stratification were present, since the heat capacity of the liquid is being effectively utilized. It is felt that the lack of axial stratification in the present case is a consequence of the larger initial ullage fraction used. This was demonstrated in results presented in Ref. 3, which showed a large stratification for small ullage fractions, even with a lower level of a/g. In Figs. 4a, b, and c the dimensionless stream furctions corresponding to the isotherms of Figs. 3a-c are plotted. Lines of constant stream function correspond to streamlines and shows the flow patterns, and the succession of the three figures indicate the transient development of the flow patterns. The table at the bottom of the figures show the range of stream functions corresponding to each digit, and are the same in the three figures. The real velocity components at each location can be calculated by Eqs. (82) and (91) for the liquid region, and by Eq. (82) and the dimensionless form of Eq. (54) for the vapor region. Increasing gradients in the stream function correspond to increasing velocities in the direction perpendicular to the gradients. It is thus noted in Figs. 4a-c that the velocities are increasing with time, and that the velocities in the liquid and vapor regions adjacent to the interface are in opposite directions. Regions of uniform stream functions completely enclosed indicate the existence of vortices, and are noted in all cases in both the liquid and vapor regions, while Fig. 4c shows a double vortex in the vapor region. Dimensional stream functions (units in ft3/sec) are plotted in Figs. 5a-d for the same conditions but at slightly different times as those in Figs. 4a-c. 2. Influence of Change of Grid Size In one case the grid size in the axial direction was changed between the liquid and vapor region, indicated in B-H 47 and B-H 53 in Table V. The sig51

CENTER AXIS co X1111 As0.097 I O 0.140 0.5 ~0.031o 0.0243- i 0.016 0;e D 4 ~~~~~~~~~~0.0 06 WALL Fig. 5. Streamlines-Run B-H 47. WA~~~~0.6 "u i~ ~~i.T temlns-u - 7

I_ ~ CENTER AXIS m 0.0243 - 0.0,55.0 485 -e -" CD I s< 0.031 0 w~~~~ 097 CM~< ~~~~~~~~~~~~0 O.062 13 9a.0775 U)~~~~/ WALL oC 0 o~~~~~z CENTER AXIS m l.0155 0.0243 ~1 O < 0.0155 0.___ __0485 _,,', 0.0315 -40.~~~~ 97~~~0.031 -.,~E — 4,,,,~097 r " bD __ 0.062 4 0~~~~~~~~~~~~~~.093 C-)~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~~~~~" 0. 17 16 ii 0.12 WALL Fig. 5. (Concluded)

nificant differences are summarized in Table VI below. The tank height is 29.2 ft, and the liquid depth is 9.4 ft. The difference in pressure rise amounts to 5% of the computed pressure rise and may be neglected in light of other present uncertainties. One difference that can be significant is in the number of computational steps required, which is related to the computer operating time. Smaller nodal spacing results in smaller time steps because of computational stability requirements, and an increase in running time of almost 50% is noted above. This may be unwarranted in terms of justifiable benefits. TABLE VI EFFECT OF AXIAL GRID SIZE System Pressure B-H No. of Nodes Nodal Spacing, ft Maximum Number _ __ After 4600 sec, No. Liquid Vapor Liquid Vapor lbf/in.2 of Time Steps 47 20 l0 0.47 1.98 18.38 1644 53 10 20 o.94 0.99 18.70 1183 3. Influence of Heat Flux Distribution The influence of the distribution of the heat flux on the pressure rise is shown in Fig. 6 for two materials, where the total heat capacities of the wall differ by a. factor of 5. For a given material, the slopes of the pressuretime curve, giving the rate of pressure rise, become asymptotically the same, regardless of the heat flux distribution. This might be expected since for a given material the total heat capacities are the same, and the total energy input rate is also the same. Also, the slopes for the wall with higher heat capacity are somewhat less than for the aluminum. This is also what one might expect, unless one examines the relative heat capacities of the walls and the fluids. The heat capacity of the liquid in the tank is 40,800 Btu/.F, and that of the vapor is 1400 Btu/~F. The ratio of the wall to the total fluid heat capacity is tabulated in Table VII for the different cases. From Table VII, the highest wall heat capacity, with polyurethane, is only 1% of that of the fluid contents, and only 0.2% with the aluminum used (Cp =.02). This being the case, one would hardly expect a difference in behavior to be evident, for a given heat flux. Such is not the case, as is seen in Fig. 6, where a distinct difference in slope between the two materials is seen. The explanation lies in that not all of the heat capacity of the liquid is brought into effective use, because of the time lag associated with the dynamics of the fluid motion taking place from the walls to the interior. 61

B-H Btu/ft2 -hr _EL CP PCP Symbol No. (q/A)L (q/A)v Btu/lbm.OF Btu/OF-ft2 0 52 64. 64. Poly..35.200 * 47 165.5 15.55 Poly..35.200 A 56 64. 64. Al.02.040 A 59 165.5 15.55 Al.02.040 22 21 A / 20 A A 19- A A / /A 18iU) / A %-O C ALUMINUM 17/ A D(r~~~~ ~POLYURETHANE D/ 16 I A / A 15- / A 14 - A 13- 0 1000 2000 3000 4000 5000 6000 TIME ( SEC ) Fig. 6. Effect of heat flux distribution on pressure rise. 62

TABLE VII COMPARISON OF HEAT CAPACITIES OF TANK AND CONTTENTS-COMPUTER MODEL Material Polyurethane Aluminum Specific heat, Cp,Btu/lbm- ~OF.35.2.02.002 Wall heat capacity, pCp8, Btu/~F-ft2.200.400.040.004 Total wall heat capacity, Btu/OF 400 800 80 8 Ratio -total wall to fluid heat capacity 0.95% 1.9% 0.19% 0.019% With the polyurethane, the pressure rise rate is more rapid, initially, with the higher heat flux to the liquid. Since the total energy input rate is the same in both cases, it appears that heat transfer to the liquid has the primary influence on the pressure rise. This is reasonable on physical grounds. The system pressure is directly related to the liquid-vapor interface temperature, and the pressure rise rate is greatest when the transport rate of energy from the container walls to the interface is the greatest. In light of the existing axial body forces, this is the case with greater heat transfer to the liquid region. Examination of the computed temperature distributions shows that the liquid temperatures are greater for B-H 47 than for B-H 52 throughout. With the aluminum walls of low heat capacity, the initial pressure rise rate is more rapid with the low heat flux to the liquid, opposite to that noted above, although the difference between the two heat fluxes is not so great. Detailed examination of the computer output results shows that this apparent reversal of the effect of the heat flux distribution is a consequence of the very low heat capacity of the aluminum wall coupled with the provision in the program to limit the superheat of the wall, as designated by the "incipient boiling point," ATwmax. With the high heat flux to the liquid, the wall temperature rises very rapidly to this point. Any further rise above this level (which is imposed as constant above the instantaneous saturation temperature) is suppressed by evaporation of the appropriate amount of liquid from the vicinity of the wall directly into the ullage space. Because of the large ullage fraction this in itself has negligible effect on the pressure in the ullage space. Because of the continual transport of this excess energy to the ullage, by way of the latent heat, the energy available for heating the bulk liquid is less, even though the heat flux to the outside of the walls in the liquid region is greater. Hence the pressure rise is less. In a real situation, the vapor generated by boiling at the wall would pass to the. ullage space by motion through the liquid, and in so doing would be transferring part of the latent heat energy to the liquid, and stir the liquid additionally. This would serve to increase the rate of pressure rise for a given heat flux. Insofar as the present model does not take into account the transfer of heat between the vapor bubbles and the bulk liquid, nor the stirring action near the wall induced by the bubbles, the model

must be considered incomplete. 4,. Influence of Wall Material For the same total energy input rate, the influence of the container wall on the pressure rise is indicated in Fig. 7. For convenience, the wall properties of specific heat and total heat capacity are included. The stainless steel wall has the largest total wall heat capacity and the lowest pressure rise rate. One notes in Fig. 7 that for the same heat flux distribution, equal in the liquid and vapor regions, that the pressure rise rate increases directly as the total wall heat capacity decreases successively with Runs B-H 55, 57, 52, and 56. A zero wall heat capacity would correspond to the case where the imposed heat flux is applied directly to the fluid contents. In these four runs the wall thickness and material vary, hence the possible compound influence of axial conduction is present. For the same material, wall thickness, and equal heat flux, one can observe the effect of the specific heat of the wall, Cp, alone in Runs B-H 56 and 57, with the same axial conduction in both cases. A very large increase in pressure rise rate occurs, and were the heat capacity decreased further, further increases in pressure rise would take place. This was done for one case with the heat flux being higher to the liquid than to the vapor, Runs B-H 59 and 63 in Fig. 7. An operational difficulty arises as the wall heat capacity decreases, which in effect increases the net heat flux to the fluid. Owing to the larger temperature differences in the fluid the velocity increases, requiring smaller computational time steps because of the stability limitations imposed by Eqs0 (131)(134). The result is an increase in the ratio of computer time to real time. As an example, in Fig. 7, Run B-H 63 required approximately 20 min of computer tine to cover 10 min of real time, while Run B-H 59 required 20 min of computer time for 100 min of real time. 5. Influence of Body Forces The influence of changing the axial body force on the pressure rise is shown in the lower part of Fig. 8, for one set of conditions of high liquid heat flux with the aluminum walls. The effect of the different body forces is almost negligible for this case, and arises because of the predominant role that the process of bulk boiling plays in suppressing the wall temperature rise for the particular conditions selected. The vapor generation taking place at the liquid vapor interface is negligibly small compared to that associated with bulk boiling. Changing the body forces should have no direct effect on the vapor generation of bulk boiling, but should influence the phase change taking place at the liquid-vapor interface since body forces affect the natural convec64

Symbol B-H Btu/ft2-hr PL C pCpE No. (q/A)L (q/A)v Btu/lbm-_F Btu/OF-ft2 56 64. 64. Al.02.04 o 57 64. 64. Al.2.4 o 52 64. 64. Poly..35.2 A 55 64. 64. s.s..11.63 + 59 165.5 15.55 Al.02.04 22- x 63 15.55 Al.002.004 21 + + 20 + / + 19 /+ / 18 / + _'AL en__ / Cp.02 a. 0 -17+ // 3 pA:.02A p + + / //13 SS /+ + 0 15 + / C =.002 14 0 1000 2000 3000 4000 5000 6000 /+ /- I s.s 13 /-,o 12 I I I I I 0 1000 2000 3000 4000 5000 6000 TIME (SEC) Fig. 7. Effect of wall material on pressure rise.

.2 a/g = 1.0410' w.15> 1 r C 0 6 2.5x104 (q /A)v15.55 BTU/HRFT2 I6 a 0 10 - 20 (q/A)1v15.55 BTU/HR-FT2 0 16w 14~ ~66 a. 1312 0 I000 2000 2600 66

tive process by which heated liquid is brought to the vicinity of the interface. That this is indeed the case is noted from the upper part of Fig. 8, which shows that increasing the body force increases the rate of evaporation of the liquid. Were the proper combination of parameters selected such that bulk boiling did not occur, it could then be expected that a more pronounced effect of a/g on the pressure rise would be present. 6. Influence of Ullage Fraction Figure 9 indicates that changing the ullage fraction from a value of 2/3 to 1/3 has negligible influence on the pressure rise. The flux to the liquid and vapor regions are made equal in order that the total energy input rate be the same for both cases. Even though the pressure is changed negligibly, a significant difference in the mass evaporated exists, with a larger mass evaporated associated with the larger ullage fraction. The trends observed are similar to that obtained with liquid oxygen (Ref. 3)- Were the heat flux to the liquid to be different from that to the vapor, then differences in pressure rise rate would occur with different values of ullage fraction, since the total energy input rates would also differ. Also, differences in behavior can be expected if the ullage fractions are very large or very small, for the same heat flux. The physical explanation as to why the pressures do not change for the present case in Fig. 9 lie in counteracting effects. As the ullage fraction decreases, less vapor is needed for a given pressure rise, hence less energy is stored as latent heat. However, the increased heat capacity associated with increased amounts of liquid compensates approximately for this. E. DISCUSSION The pressure and total mass evaporated from computer Run B-H 47 are repeated in Fig. 10 as a function of time. The program listing in Appendix B incorporates the ability to sotre and/or punch intermediate results for reinsertion at a later time to continue the computations. In the case where the results are punched, only distributions of the stream functions and temperature are retained, along with the system pressure (i.e., the U and V velocity components are discarded). To observe the influence of not using the velocities at the start of each section, computer Run B-H 47 was repeated by starting and stopping the computational process a number of times, as indicated at the bottom of Fig. 10. The general overall trend is affected only in a minor way if one neglects the "starting transients," and the comparison with the continuous run is quite good. Included in Fig. 10 is the final measured pressure in the LH2 tank on the AS-203 flight, as taken from Ref. 17, and is considerably higher than the results predicted by the particular computer model included. A number of factors which contribute to this difference have already been discussed, and the more significant of these will be reviewed here. 67

Sb s a B -H N. Ullage /A PrEva No. Fractio Bft2-hr Material 21 o 56.667 4 6 6 ~ 6 764. Aluminum ~5 64. Aluminum 9 20 -8 19 _Jr U) — 18 c3 U) a~~~~~~~~-6 -0 14 -5U "-J~ ~ ~ ~ ~ ~ ~ ~~~~~~~~~~~~--- oFI~ 0 14 0 2 0 ~IOQO 20QC 3OO! 400Q 0Q~ 6 40005000 6000 TIME (SECONDS) Fig. 9. Effect of Uliage fraction on pressure rise. 68

40 / Final Measured - Pressure - AS-203 o Pressure - Piece wise Run * Pressure - Continuous Run Computer 30 A I Total Mass Evaporated -Piece Wise Run BRun * Tota I Mass Evaporated - Continuous Run H 47 O Final Measured Pressure-AS-203 Flight (Ref.17) O Calculated Total Mass Evaporated-AS-203Flight(Ref. 17) 25t /0 -300 / / / / / CL 00 w 20/ - 200 crr/ o 15+ ^4Totl" 1M50 Rn w / ~~~~~~0 ~A0 00 Computer Calculated Pressure Aa Rise. Total Energy Input Rate > to Outside of Tank=35.8Btu/Sec., 0 ]lot~ 0- -I004 0000 T0 Ag A, ComEvaporated 00 0 10 00 2000 3000 4000 5000 000 Ist. Run 2nd.Run 3rd. Run i- 4th. Run t Run TIME (SEC) Fig. 10. Composite computed pressure rise. 69

The geometrical difference between the actual system and the model is obvious. Although the total volumes and total energy input rates were maintained the same, the wall heat flux could not be modeled simultaneously because of the geometrical differences in wetted and nonwetted heat transfer surface areas. These differences are shown in Table VIII. Cases A and E correspond to the heat fluxes for the actual and computer models listed in Table IV, each case having the same respective energy input rates to the liquid and vapor regions. For case F, the heat flux is considered uniform over the entire actual tank surface area. The heat flux for this case is almost one-half that of the smallest used for the computer model. In all cases the total energy input rate is 35.8 Btu/sec. If the heat flux for the computer model were maintained the same as that based on the actual geometry, then the total energy input rates would differ between the two cases. TABLE VIII COMPARISONS BETWEEN ACTUAL AND COMPUTER MODEL HEAT FLUXES Computer Model, Btu/hr-ft2 Based on Actual Geometry, Btu/hr-ft Case (q/A)L (q/A)v ~/A L (c/A)v ~ Wetted Unwetted A 165.5 15.6 74 10.9 E 64 64 28.7 44.7 F 37.8 37.8 The greater heat flux to the wall in the liquid region, in the computer model, results in a large rate of increase in wall temperature. The combination of conduction and convection in the liquid is not sufficient to transport the energy away from the wall, and the incipient boiling point is quickly reached. In the results presented thus far, the incipient boiling point was included as an input parameter. Since in the physical process of nucleate boiling changes in the wall temperatures are relatively small, this was simulated by suppressing the wall temperature with an appropriate release of vapor to the ullage space, computed with Eq. (135). The generation of this vapor in effect reduces the rate of temperature rise of the bulk liquid that might otherwise take place, and with a large ullage fraction the pressure rise rate would also be smaller. For one case only, a computer run was made in which the suppression of the liquid wall temperatures did not take place (i.e., no nucleate boiling was permitted). The resulting system pressure behavior is plotted in Fig. 11 as B-H 71. Included for reference purposes, as a basis for comparison is the otherwise identical case B-H 56. Computations were carried out for only 2000 sec, which corresponds to 20 min of computer time. However, if the pressure behavior in Run B-H 71 were extrapolated linearly to 5500 sec, the predicted 70

WALL MATERIAL: ALUMINUM WALL THICKNESS:.01165 FT UNIFORM q/A = 64 BTU/FT 2HR. 19 Cpw -.02 BTU/LBM- ~F a/g = 1.7 x 10-4 TSymblol B-H Nucleate 8 ybol No.'Boiling 0 56 Yes CI Ad 71 No 17 a) c 16 C) C, w 1 14 13 12 0 1000 2000 TIME (SEC) Fig. 11. Effect of nucleate boiling on pressure rise.

pressure would be 38.6 lbf/in.2, as compared with 37.7 lbf/in.2 in the actual case (Fig. 10). Although the pressure comparison is quite good, the computed bulk liquid temperatures become unrealistically high. For Run B-H 56, with nucleate boiling, at the end of 2000 sec the saturation temperature was 38.20R with a mean bulk temperature of 38.80R, whereas for Run B-H 71 the saturation temperature was 39.4~R with a mean bulk temperature of 387.90R. This is a result of the high heat flux to the liquid, case E in Table VIII, and in effect states that, in a real system patterned after this computer model, nucleate boiling would most certainly occur at the wallo In Section VII-A of Ref. 17, statements are made that nucleate boiling indeed occurred on the AS-203 flight, with the vapor apparently recondensing in the bulk liquid. The computer model used here considers that the vapor generated by nucleate boiling at the wall passes directly to the ullage space. Vapor bubbles traveling from the vicinity of the wall to the ullage space would in fact introduce two effects, both of which must be included in the computer model in order to have a more complete description. The motion of the bubbles would tend to increase the velocity of the liquid, and also would tend to heat the liquid as it travels toward the liquid-vapor interface, if collapse takes place. The effect of each of these would be to increase further the rate of pressure rise. In the course of the transient processes occurring in the closed container, both with and without simulated nucleate boiling, it was observed that simultaneous evaporation and condensation occurs at the liquid-vapor interface during certain early intervals. This is demonstrated in Fig. 12, which includes the same two cases presented in Fig. 11. The ordinate is the local relative mass flux, either evaporation or condensation, at the liquid-vapor interface as a function of radius and time. The logarithmic ordinate scale increases in both the evaporation and condensation directions. With nucleate boiling, Run B-H 56, condensation near the centerline persists for a considerable period. Since the boiling adds a great deal of vapor to the ullage and the bulk liquid temperature rise is relatively slow, the compression of the ullage vapor tends to increase the condensation and reduces the tendency for boiling as the system pressure increases, which is related to the temperature at the liquid-vapor interface. The evaporation rate near the side wall of the tank quickly reaches a steady value, which indicates that the temperature gradients in the liquid and vapor in that region remain constant. With no nucleate boiling permitted, Run B-H 71, condensation also occurs near the centerline, but for a short period only. The evaporation rate becomes quite large, due to the establishment of large temperature gradients in the liquid, which is possible when no restriction on the level of wall temperature is present. 72

'0-I B-H Time (sec) Nucleate.05 No. 200 400 6oo00 1000 Boiling O 56 0 A 0 O Yes 710 A U No D<_ U) LL z U) z 10 Ld Cr > 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 t~~~~~,% o_~~~~ A CO, ~ 6 %, - 01 2IS1 51 71 92 Z 0.05 t -J 10w.5 SIDE WALL. 5 C I0['CENTERLINE 50 109- c~ i0'~k~ ~ I 2 3 4 5 6 7 8 9 1011 12 13 14 15.16 17 18 19 2021 RADIAL NODAL STATIONS Fig. 12. Effect of nucleate boiling on local liquid-vapor interface mass transfer. 73

An additional improvement in the computer model could be made by the use of nonlumped walls, as discussed earlier in Sections C.1 and C.2. In the actual case, aluminum is on the outer side with an insulating material on the inner side. This tends to make the temperature on the outer side more uniform because of axial conduction. The boundary condition of an imposed heat flux can result in somewhat artificial results if the axial conduction is not included, and is particularly important in the vicinity of the liquid-vapor interface where the axial gradients in the wall can become large. One additional observation on the numerical procedure should be made. On beginning the computational process, the changes take place from the region where the disturbance takes place, and progress one node per computational time step. This means that one must not attribute undue significance to the results until sufficient computational steps have taken place to influence all of the nodes. This is particularly important when one is attempting to model a physical system which itself is not stagnant at the outset by a system initially at a uniformly stationary condition. 74

APPENDIX A COMPUTER PROGRAM FLOW CHART 75

T READ N= WRIE/IA, PRESS, T(l,J), 5F(I,J) ETC- T J;01 READ AND CONST,CONSTDE=1 WRITE INPUT TSAT=" TINIT= —- CONST 2, CONST 3, OD PR, PRV F W(I,N +1) W(,)f[FIJjISF(M +I,J)=o {<3 | ~~~~W(l,J)=f [SF(I,J)]|9 L. Wall LL.-V. int. CODEO BEITA( HFG NT=N02 PlPRESS U(l' 1) =~ 00~,J)=~~t W(I,N+ - ~~-~ U ~~ 3 F(MAJ)=~~~- U(I,N)-~ I U(1,2)=~~~I UI.'~~I-I VM2J= ~ -- GJZ~ T \ V. V. int. L.-V. int. V. V. V. V. int. SMAX>~ S... SMAX=O C) 1(PEl'J) W(H U WU) =... vA V J) T L.+V. V. V. V. F BOO=... FI SMAX=S >1 BO =... S < TSMX< SMAX= S DT=.8/S MAX F = BOO X= an X=S S=~~~ SMAX < SMA BO DT>.8 I,~~~~~ ~~~~~~~~~~~~~F DTDXL(J) ~~- T0(IJ)=T(I,J) WO(I,J)=W(I,J) 1-~R(,)=~~)-f TPt,N+l)= ~ T(MclNcl=1 ~t-I,N-4 (1,N -~~5- T(i,N+l) TIME=-~ Entire Region Entire Region Entire Region Top Surface int. L. DTDXG(J) ~ - UL(J) = -I UU=UU DT' DDRW =... QWI~ DQIL I QI DQWDT= ~ RQIN,.+V. V. Wall T(M-ti,J) T~I SAT-f(PRESS) Pl P-PRESS H — PRESS::~~... M = ~ 1 DMT H~'~- DME H ~ - DQIDT = ~ H- DQILDT =. TOM z~~ -lT(,)=~ T(ld)= ~ I- T(Id) H w(,- W~,J)=~~~.-H.SF(Mr+ldJ)=O - NE1=0 2) — NE1=NEI+l' @ - 4 TMAX RMAX NE1> T(I,N-c1 )=T(M+IN)+DTW*CONST1 RR3=~~ SF(I,J) N N >EPSL _ < FR >(T(M iN3 FI I L +DTSW T 0 "CONST F T(I,N)=T(M+I,N) + ~ ----- DQS"=0 — D Q L —- 0 MB=O -- DQL=Ot- DQI =... — DMB <0 ~t — M~ > +,N) -4- DTS* DMB- 0 CONST 1 SF(M+1,J)=f(RO(M+1I JJ) W(I,J)=~~~H T(I,J)=.... T(I,1=~. ) (P+I,J)= ~ HT(P I'l) - ~~H UG= DWDT=... H uu=uu- - - DMTT=~~ ~ DMBT= - ~ ~ F RR3= ~~C S F I(J) SF(M+ 1,J) SFOIN44F (MM+ 1,N~l NE2=0 C 7- - NE2=NE2+1 2> E N E T= 1 AND RAX NE2=1 NLO SF(I,J) ~ U(.IN. ~~ U(1,2)= ~ U(I'l)~~ U0I,N) 60~ U(Ilj)=L 0,J) O(I,=)V(I,J) 103 w,J)= ~ W(P+I,J) W(I,N-tl)~~ V, L. L. ~~ ~ ~~~~~~~~~~~~~~~V. V. V.+L..

APPENDIX B LH2 COMPUTER PROGRAM LISTING 79

FORTRAN IV GCOMPILER MAIN 12-12-68 18:56.22 PAGE 0001 C PROGRAM FOR CYLINDRICAL CONTAINER FOR IDEAL GAS-HYDROGEN 0001 DIMENSION U(31,21)tV(31,21),T(31,21)tSF(31,21),W(3-,21)'T031- 21,IWO( 31,21),SF I(31,EJ (31),F(31),01( 31),DRI (31 ),DR2(31,) DR3( 31), 2DR4(31),R1I (31) R2 (31),R3 (3 1),R4 ( 31),R5 (31.-)-,-A — 1(31'1 i A2-(31),-A3 ( 31),-... 3233C2(31) C 1 (31,21),C3(131,21) E,J1 ( 31) 01 ( 31),D02(31),R6( 31),DR6(31), 4UL ( 31),DR5(31'), DR7 ( 31), ODR-8( 31 ), A4(31), VG( 31'). 5R0(31,21),DOTODRW(31 ),DTDXG(31 ),UO(31,21 ),VO(31,21), DTDXL( 31) HL(31 ) 6,HG(31 3 ),C4( 31,21), RT (3'1,21), ZTT(-12),FHHFG( 12 l.,TT(12),HHFG(12) 0002 INTEGER CODEHPTAUJl,TAU2 0003 REAL KG, KL, M 1, M2, M3, M4,M t-5,6tMINP LNE', NEWV, NUG, NULM 11, M2 3 NP2 0004 204 FORMAT(3FI12.4,F12.8, 4IFl2'.8,,_E,_ 6.8/2E13. 6/_5E16.8) ) 0005 NAMEL I ST/DOAT A/A AOVERB P, MN,X CODE DT,NMAX PO, Gt QSURFG QSURFL IDTWDTS, ALPHANEWKL,iBEITA ROLCP ALPHAV, NEWV KG,CPV CV, 2RGAS,ALPHAW,C PW ROW DELTA~ TAU L,N l E E PSLON NC,NR-;t1-'[ B t- B 3B-2-4it-~ -Z, 3ZTT~FHHFG,NDIMvTAU2 ~~~~~~~~~-/A A2-/ —O'V T' — N -S7o7b;~/ TdS,-2- tC ON-STZf i',CONST 3,P'R, PRV.... 0006 101 READ(5,DATA1,END=500) 0007 WRITE{(6,DATAIY........................................... 0008 KI=M+I1 0009 K2=N+ 1 0010 K3=P+1 0011 K4 — 0012 K5=N-1.. 0013 K6=M+3 0014 K7=P-1 " -00-15 K8=M- 1 0016 DATA AI~A2,A3,A4,C2,OlD2, I',DRiDR2,I)R3,DR4,DR5,DR6,DR7,DR8,'DTDRWDTDXGDTDXLEJEJ'lFHG',HL'RR2,R3,R4,R5,R6,SFIUL,-VG/ 21023*0.0/,CLC3,C4,ROSF,TTO,UUOV,VO,WWORT/9114*0.O/, 3BO~F-,0- F' —-,' E, F T3 i TIME" R-i, R E2,E3'UU TAUP l, 1' DMT, DMBT,DM' A6, M6,NP',. 4DMBDMTT/20*0.0/NNT NE1,NN E2,N2/40/ 0017 I1=M/2.+1 0018 Y=AOVERB 0019 B=A/AOVERB 0020 PR=NEW/ALPHA 0021 PRVNEWV/ALPHAV 0022 RNEW=NEWV/NEW 0023 RALPHA=ALPHAV/ALPHA 0024 DR=I.O/N 0025 ER2=DR*DR - 0026 GAM=CPV/CV 0027 -DYY=20ER 0028 ER6=ER2/2.0 0029 PRESS=PO 0030 BB=(B3*B4-ALOGIO (PRESS) +B1) 0031 C =(82-B3*ALOGIO (PRESS)+B1*83) 0032 D=BB*BB -4.*84*C 0033 TSAT-=(-BB+SQRT(D))'/(2. *B4) 0034 TINIT=TSAT 0035 CONST = A*A/ALPHA 0036 CONST1=Y*PR*G*BEITA*(A** 3)/NEW/NEW - - Ci00373 — - CON-ST2= 1.O/CONST 0038 CONST3=1.0/CONST1 0c39 WRITE(6,DATA2) 0040. MI=AOVERB*QSURFL*(.A** 5 ) *BEIT A*G/ALPHA/ALPHA/NEW/ROW/CPW 80

FORTRAN IV G COMPILER MAIN 12-12-68 18:56.22 PAGE 0002 1 /DELTA 0041 M2=KL*A/ALPHA/ROW/CPW/DELTA 0042 -M3= KG*A/AL PHAROW/CPWPW DELTA 0043 M7=ACVERB*QSURFG*(A** 5 )*BE ITA*G/ALP PHA/AL PHA/NW/ROW/CPW./DELTA 0044 M5=CGNSTI*TINIT - 0045 D2R=DR/2.0 ~ 0046 AOVRB2=AOVERB*AOVERB 0047 00 11 J=2,K2 0048 Al(J)=ER2*ER2*(J-1.)*(J-1.)0049 -C2(J)=1.+I./(2.*(J-1.)) 0050 A4(J)=1.0/(J-1.0)/DR/4.0 0051 11 R(J)=12*( J-1.O)*ER2 0052 IF(CODE.NE.1) GO TO 14 0053 READ (5, 2C4)A,PRESS,PO,X, NTTIMEUU,DMBT,DMTT,((T(I,J),J=1-,K2), 1I-,K3), {SF( I, J),J=K2), I=1,K 3 ) 0054 DO 13 I=1,K3 0055 0013 3=1,K2 0056 13 TO( I,J)=T(l,J) 0057 - 14 N3=1 0058 N4=l 0059 Pl=PRESS 0060 102 NT=NT+ 1 00-61 TSATK=TSAT/1.8 0062 CALL ATSM( TSATKZTT.,FHHFG, 12,,TTHHFGNDIM 0063 CALL ALI(TSATKTTHHFGHFGHGJNDIMO.1E.R) 0064 HFG=0.215*HFGJ 006 5 RP=PRE S S/PO 0066 [=(P+M)/2.+l 0067 J=N/2.+1 0068 BEITAG = 1./I(TINIT+T(I,J)/CONST1) 0069 M4=(GAM-1)/GAM OC70 CT=I. 0071 DX=X/M 0072 DX2=DX*DX 0073 DZ=(I.O-X)/IP M) 0074 DZX=DZ/DX 0075 DZXi=DZX+1. 0076 DZX2=. 5*DX*DZ*DZX1 0'0777 - -. — -.-.-.-. —.-.D DZXH=2./ ( D Z+) - - 0078 DZ2=DZ*DZ 0079 DRZ=ER2/DZ2 0080 F3=DRZ*Y*Y 008 BJ1=2. * ( 1+F3) 0082 DXR=DX2/ER2 0083 DRX=ER2/DX2 0084 F1=DRX*Y*Y TdY085 F2= DXRJRY/YY -- -- -- - OC86 02Z=DZ/2. 0087 ZOVR2=Z 2.0 0088 _aBJ=2.* (1.+F ) 0089 B 0 —= —-- 2S * P R * Y* Y-DX'2+2-P RE-R-2I 7E. -- 0090 EJ(1) 0. -00 l — EJ1(1)=0.OC92 DO 15 J=2,K2

FORTRAN IV G6 COMPILER MAIN 12-12-68 18:56.22 PAGE 0003 0093 A2(J)=DX2*ER2*(J-. )*(J-1. /Y/Y 0094 A3( J)=DX2/Y/Y/(J-1. /ER2/2. 0095 DI(J)=BJ-C2(J)*EJ(J-1) 0096 R2( J)=12*(J-1. ) *DR*DX 0097 D2( J)=BJI-:C2( J )*EJ1 (J-1 ) 0098 EJ1i (J)=(1-1/(2.0*(J-1.0 ) )/DZ(J) 0099 R4(J) =Y*Y/(J-t. ) / (J-1. )/ER 2/0X2/2. -0100 R5( J )=Y*Y/(J-1..) / (J-1.)/ER2/DZ2/2. 0101 R6( J )=12*( J-1. ) *DR*DZ 0102 15 E J(J )=-l.oJ-1 2*1-(' -. - 0}/D1( J) 0103 -F(-CODE.NE.1 ) GO TO 107 0104 DO 16 J=2tN 0105 16 SF(M+I,J)=0. 0106 DO 17 1=2,M 0107 DO 17 J=2,N 0108 17 W(IJ)=((SF(1+- J)-2 *SF(IJ)+SF(I-,J} )*Y*Y/DX2-(SF(I,J+1)1 SF ( I,J- ) ))/ER2/(J-1.0 )/2+( SF( I,J+1 )-2.O*SF (I,J)+SF( I,J-1) )/ 2 ER-2)-IER2I(-J-1i.-O {-T('-'-i-;-O )/1 J —... -.-.-.-.-.-. —. 0109 D0018 I=2,M 011'0 18 W(IN+1)= (8.*SF(I,N)-SFXF(IN-1)-7-.*SF(I,-NI+~-l)DYY -.-. —-.0111 DO 19 I=K1,K3 OTV2 - 0019tS J~lwK2 - - -- - ------ - - -------------- ----- - 01'13 19 RO( I(I,J)=CONSTl.*PRESS*144./RGAS/(T ( I,J+M5) 0114 D00 20 I=K4,P 0115 20 W(I,N+1 )=( 8*SF(I,N-)-SF(I,N- )-7*SF(I,N+1))/DYY/RO(I,N+I)l 0116 UG=-UU*( ROL/RO( M+1, 1 -1. 0) 0117 DO 21 J=2,K2 00118 21 S F ( M+:1,',J-)-,R C(M+1, l"'J )'*-U G*ER"2*'( J'- 1.0 )'-* ( J- 1'.0 ) /2. 0 0119 DO 22 I=K1,P 0120 U( I,N)=-2.*(3*SF( I,N- 6*SF( I,N- )+SF ( I N-2) )/R1(N)/RO( I,N) 0121 U(I,1=SF ( I 2 )/ER6/RO(,t 1 ) 0122 22 U(I2 )=2.*(6.*SF(I,3) -SF (,4-3.*SF(, 2 ) )/R1(2)/RO( I,2 0123 DO' 23 I=K1,P 0124 DO 23 J=3,K — S 0125 23 U(I,J)=(SF( I,J-2)-8*SF( I,J-1)+8*SF( ItJ+1 -SF(IJ-2 ) )/R1(J) 1 /RO(IJ) 0126 DO 24 J=2,N 0127 VM+ 2,J )=2.*( 3.*SF ( M+2,J )+SF ( M+4, J )-6.*SF (.M+3,J )+2.*SF( M+1,J) I )/R6(J)/RO(M+2 J ) 012-8 V —— G-()-=-'-"-2-.-*'('SF'(M+3,"-J) —-8-.,SE ( M +-,-J) +7-, *SF (M+1, J-) /R 63 (J)/RO(M+-1,J 1 ) 0130 DO 25 J=2 N 0131 - DO 25 I=K6,P 0132 25 V(IJ)=(SF(1 +2,J)-8*SF(I+1,J)+8*SF(I-1,J)-SF(I-2,J) )/RI(J)/RO. 0133 DO 26 I=K4,P 0134 DO 26 J=2tN 0135 26 W(I,J)=I(.SF(I+1,J)-2.*SF(ItJ)+SF( I-1,J)_*Y*Y/DZ2-(SF(I,J+I)1 SF( I,J-1) )/ER2/(J-1.)/12+(SF(I,J+1)-2*SF(I,J)+SF(I,J-1)i/ER2)/ 2 ER2/(J-1.)/(J-. )/RO( I,J)-(U( I,J)*(RO(I,J+1)3-RD I, J- i —--- D — -i-102 -T: -R(, J 1 —-— R- -l,)-/bZ'i-A4( J)-/ARO'[ J 4 ----- 0 260 ---- - --- ------------- ---- -, — - 0137 WIP+1,J={-F(-1SF _P-1,J)+8*SF(PJ)-7.*SFi(P+I,J3) ~_ 82

FORTRAN IV G CQMPILER MAIN 12-12-68.18:56.22 PAGE 0004 1 *RS(J)/RO(P+1,J)0138 260 W(1 J)=(-SF(3,J )+8*SF2,J } )*R4(J ) 0139 CODE=2 0140 GO TO 103 0141 107 SMAX=O. 0142 00 27 I=2,K10143 DO 27 J=1,N 0144 S=80+ABS (U( I,J) )/DX+ABS( V I,J) ) /DR 0145 27 IF( SMAX.LT. S) SMAX=S 0146 IF(PRV.LE.1.) GO TO 28 0147 BOO=4.PRV/ER2+2. *PRV*Y*Y/DZ2 0148 BO=2. *-PRV* ( Y*Y/0DZ2+ 1./R 2- 0149 GO TO 301. 0150 28 B00=4. /ER2+ 2. *Y*Y/ DZ2 0151 B0=2.* ( Y*Y/DZ2+1./ER2 ) 0'152 - 301 DO 29 I-K4 P 0153 DO 29 J=2,N 0154 S= BO+AB 8S(- -i —J)''/D Z+ABS( V( iJ )/ DR 0155 29 IF(SMAX.LT. S) SMAX=S 0156 DO 30 I=K4,P 0157 S=80O+ABS(U( 1,1 ) )/DZ 0158 -— 3-0 -- F-USMAX-.'LT.S —) S.MAX —= S - -- -- 0159 IF ( SMAX*DT).GT.0.8) DT=O. 8/SMAX 0160 T MET — M=TIE+DT 0161 TAU=TIME * CONST -''01i-62 X- =X+UU*DT 0163 DX1=DT/DX 0164 -- - OZ1=DT/D-Z 0165 DZ3=CT*DT*Y*Y*R ALPHA/ DZ2 0166 Z5-=2.*DZ3 0167 DZ7 = DT *Y * Y PR * RNEW / DZ2 01b8 - DZ8=OTT*Y*Y*ALPHAW/ALPHA/DZ2 0169 OZ88=OT*Y*Y*ALPHAWA LPHAW/ALPHA DZX2 01 70 M1 =DT*.5*B EI TA*G*( IA**''6')-/( ALP HA*ALPHA*NEW *ROW*CPW*DEL TA*B ) 0171 M23=. 5*T*A/( ALPHA*ROW.*CPW*DELTA) 0172 - - -- - - -Y3-C-T —'DT*R AL'-HA/ ER2 0173 DY1=2..*DY3 01-74 DY7 =. DT * RNEW * PR 7"-ER2 0175 DY5=4.*DY3 0'176 - - - - --. —- --- X — - - -- -— D 3 —DT' *Y-"/*Y D -X2 --- - - - - 0177 ER3=N*N*DT 0178.4-= —DT*.. 0179 ER5=4.*ER3 0180 DX5=2.*DX3 0181 ERI=2*ER3 01 82 - -- - ----- ----— DX7T —DX3*PR - ------- 0183 DX8=DT**Y* PHAW/ALHAWALPHA/DX2 0184 ER7=ER:3*PR 0185 EI=1. 0-DX5-ER5 0186 E 2=1.0-DX5-ER1 0187 E3=1.0-2.0*DX7-2.0,*ER7..0188 0 3I JI - 0189 F I=0.5/( J-1.0 ) — 0-190 s c — -- - - -' — - J- E- - U — 1. _y 0191 DR(J}) =ER3.*( 1.0-FI) 83

FORTRAN IV G COMPILER MAIN 12-12-68 18:56.22 PAGE 0005 0192 DR2( J )=ER3*(1.O+FI I 0193 DR3 (J)=ER3*( 1 O-EI ) 0194 0 DR4( J )=ER3*(.0O+E I) 0195- R3( J)=PR*ER3/(J-1.0)/2.0 0196 D R5 ( ) = CT*RALPHA*DR 1( J ) 0197 DR6(J)=CT*RALPHA*DR2(J) 0198 DR7(J), = DR3(J) * PR * RNEW 0199 31 DR8(J) = DR4(J) * PR * RNEW 0200 D00 32 I=2tM 0201 DO 32 J=lN 0202 C(II,'J)=U(,J)*DX1 0203 32 C3( I-,-)V(IJ) *ER4 0204 T ( 1,N+I)=TO( 1,N+ ) +DT*MI-DT*M2*(11*TO(1,N+I )-18.*TO( 1,N)+9.* 1TO(1,N-1)-2.*T( 1,N-2))/6./DR +2.*DX8*(TO(2,N+1)-TO(IN+'1) 0205 00DO 33 I=2,M 0206 -3-3 TfN(I-;-(Nii)-=t I fN+ Di)t'T*Ml-DT*M2*( 11.*TO { I, N+1 )-18. *T0 ( I, N)+9.*TO( I, 1N-1)-2.*TO(I,N-2 ))/ 6./DR+DX8*(TO( I+IN+I)-2.*TO(I N+1)+TO(I-ltN+ 0207 T ( M+ 1 N+ 1) =T 0O( M+ 1,N+1 ) +DZXH* ( ( SURFL*DX +QSUR FG*DZ )* M 1 - ( K L* DX+KG* lDZ)*M23* ( 11. *TO (Ml,It N+l)-1 8.*TOTf( IM+t I +9-'-.-TO ( M+' 1'- N- 1)-2. *TO ( M+, 2N-2) )/6./DR +Z88* ( TO( M+2,N+1 ) -DZX*TO(+ 1, N+ 1 ) +DZX*TO (MN+ 1 ) )*: ( DX 0208 DO 34 I=K4,P 0209 34 T ('I,-N+I-) —-; TO(I,"N+1 )+'DT*M7-DT*M3* ('1.*TO ( I, N+ ) -1. *TO( { I,N)+9.*TO( I, 1N-1)-2.*TO( I,N-2)) }/6./DR+DZ8*(TO( I+l,N+L )-2.*TO(I, N+1 )+TO I-1,N+l 2)) 0210 T _ T(P+1,N+I.)=TO(P+1N+L )+DT*M7-DT*M3*( 11.*TO( P+1,N+)-18.*TO( P+ 1,N)+ 19.*TO(P+1tN-1)-2.*TO(P+1,N-2))/6./DR+2.*DZ8*(TO(PN+1)-TO(P+1,N+1) 2) 0211 00 35 J=1,K2 0212 DO 35 J=1,K2..................... 0213 WO(I;J)=W(I,J) 0214' 35 TO( IJ=T(I,J) 0215 DO 4Q2 I-K1,K3 0216 D00 402 J=1,K2 0217 402 RO ( I, J )=C.ONSTI*PRES S 144.0/RGAS /(T( I,J)+M5) 0218 DO 36 J=1,N 0219 DTDXL (J)=('*T(M+',J)-18*T ( M,J+9*T(M-,J)-2*T(M- 2J ) )/6/DX 0220 DTDXG(J)=(-*T(M+1 tJ )+18*T ( 2,J3 -9*T (M+3,J)+2*T(M+4,J) )/DZ/ 1 6 0221 36 UL (J) =( DTDXL( J)-DTDXG (J)*KG/KL)*CP *NEW*NEW*AOVERB/PR/ --- - — 1 - BEITA/G/HFG/( A** 3) 0222 UU=O. 02;23 DO 37 J=1,N 0224 37 UU=UU+( UL (J-+I+UL )) *(2.0 *J-1.0)*ER2/2.0 0225 DO) 38 I-=1,K3 0226 38 DTDRW(I) = ( 11*T (- I'N+1 ) -18* T( IN)+9*T( I,N —2*T(I,N-2)} }/6/DR 0227 QW= o 0228 DO 39 I=K1 P -0229 39 QW=QW+ (DTDRW( I) +DTDRW ( I + i) ) *Z/2. 0230 QI=O. 0232 DO 40 J=1,N 0 2-33- D40 -QI 5. +{D-TX —-DTDX- J+ 1. }'* 2,'0'*-J= —0.0 ) *E-R"2'J2-.0 - -'-O 0234 40 QI=QI+ (OTDX (J ) +DTDXG (J+L ):*( 2.O*J-l.0 ) *ER2/2. 0 84

FORTRAN IV G COMPILER MAIN 12-12-68 18:56.22 PAGE 0006 0'235 _DQWDT=QW*KG*6.28*NEW*ALPHA/BE I TA/G/A/A/AOVERB/AOVERB 0236 RQI N=OQWDT*AOVERB/6.2 8/A/A/QSURFG/( -X) 0237 DQILDT = 3.14*DQIL*KL*NEW*ALPHA/8IElTA/G/A/A a 28 ~~ S DQIDT=3.14*QI*KG*NE W*ALPHA-BEITA/G/A/A 0239 DME=DT*UU*ROL*(A** 3 )* 3. 4/AOVERB 024'0........ D - —'' —MT=DMT+DME 0241 DM=-DMB+DME 0242 PRESS=PRESS+RGAS*AOVER8/(3.141144.*(A**3)* ( 1.-X)l)*( (DQWDT1 DQIDT)*DT*A** 2 /(CV*ALPHA)-DM*GAM*TSAT) 0243 A6=(PRESS-PI)/PRESS 0244 M6= ( PRESS-P) /P 1 0245 PI=PRESS 0246 B=(B3*B4-ALOG10 (PRESS )+B). 0'247 C=( B2-B 3* ALOG 10 ( PRESS ) +B *B3 ) 0248 D=BB*BB -4.*84*C 0249 TSAT=(-BB+SQRT(D))/(2.*B4) 0250 AS=2.*DT*AOVERB*G*(A** 3 )/ALPHA/ALPHA 0251 00 41 J=1,N 0252 41 T(M +1 tJ )=( TSAT-T I N I T) *CONST 1 0253 T ( -, 1 )=TO( 1 )*El+TO( l2 )*DX5+TO ( 2, 1 )*ER5 0254 DO 43 I=2,M 0255'- 0'-.....IF(U(FI,1.LE ol.E0. ) GO'T042 0256 - C=(TO( I-1 1 )-TO( I,1!) *C1 (I, 1) 02-57?...... GO TO r43 0258 - 42 C=(TO( I,1)-TO(I+1,1))*CI( 1,I) 0259 43 T( I,l ) =TO( I, 1 )*E1+TO( 1 I,2 )*ER+2(TU (.I+ ",1 )"+TO([I-I l 1 ))*DX3 +C 026C DO 44 J=2,N'0261..44 T(1'-,J)=T('J)*''-' ——'E2+TO(2, J)* DX5+TO ( 1, J-1 )*DR1 ( J I1 +TO( 1,J+1 )*DR2(J ) 0262 DO 47 J2-,N 0263 00DO 47 I=2,M. 0264 IF(U(IJ).LE.0.) GO TO 45 0265 C=(TO( I-,J )-io(.I J)).*Cl(,J) 0266 GO TO 303 0267 45 C=(TO( I,J)-TO( I+1, J ) ) *C1 ( i tJ ) 0268 303 iF(V(I,J).LE.O,.) GO TO 46 0269 D=(TO(I,J-1)-TO(IJ))*C3( IJ) 027C-0. TO 47 0271 46 O=( TO( I, )-TO( l,J+1 ) *C3( I,J ) 02'72.. 47 " i'' TO'(IJ) T('I,-J) *E2+("TO'1(-I'+'iJ)+TO( i -,I J) )J*DX.3+ 1 TO( I, J-1)*DR1 ( J)+TO( I J+1 )*R2(J)+C+D 0273......00. 50 I.'2"" - 0274 DO 50 J=2,N 02'75 FI=R3{J)*('T(I,J+1)-'T( l,J-1')............ 0276 IF(U(IJ).LE.O.) GO TO 48 0277 C=(WO( I-19J)-WO(-'. "J')") *CI (I, J 0278 GO -TO 305 0279- 48 C (WIJ-WM J*C1IJ 0280 _ 305 IF(V(IJ).LE.0.O ) GO TO 49 0281 D=( { -O,-1- l)-WO( IJ))*C3( t —J 0282 GO TO 50 -— 0-283.........-. 49 ~ 0284 50 W( I-,J)=WO(,'J)-*E3+(WO(:I+ 1tJ)+WO(I-l, J))}*DX7+C+D+ IW0TT[1 W)( I, J-1 )*D R3 ( J } + WO IT-,.iT- DR'{(J')-F I.... 0285 00 51 J=2,.K2 85

FORTRAN IV G COMPILER,MA IN 12-12-68 18:56.22 PAGE 0007 0286 51 SF( M+1J)=0. 0287 NEl=O 0288 52 NE1=NEI+l 0289- IF(NE1.G-T.NE) GO TO 104 0290 DO 53 I=I,Kl 0291 DO 53 J=l, K2 0292 - 53- C3 (ItJ)=SF(I,J) 0293 I=1 0294 54 I=1+1 0295 F(1)=O. 0296 DO 55 J=2,N 029'7..D I 1 J )=- C 3 ( I +1 D J ) +C 3 I-1 i,-J } } F 1-W I J ) A 1 ( J ) 0298 55 F( J)(DI ({ J)+F (J-1) C2( J) /01(J) 0299 00 56 J=1tK5 0300 H=N+I-J 0301 56 SF(I,H)=EJ(H })*SF ( IH+ )+F(H) 0302 IF(I.LT.M) GO TO 54 0303 RMAX=C. 0304 00 57 I=2,MtNR 0306 RR3=ABS((SF( I,J)-C3( IJ) )/ (C3 (I,J)+1.OE-20) } 03-07 - - -'5-7. ——. TFF(RMAX7OLT..RR3 ) -RMAX=RR3 0308 IF(RMAX.GT.EPSLON) GO TO 52 0309 DO 58 I=lK'I 0310 IF(T( IN+I).GT.('T(M+1,N)+DTW*CONST )) T( I,N+1)=T(M+IN)+DTW*CONSTI d0311.~~-~~- ~ 58 IF(T(I-N).GT.(T( M+i,-+N)+DTS*CONST1)) T(I, N)=T(M+ l N)+DTS*CONS T 0312 DQS = O. 0313 DQL = 0. 0314 OMB=0. 03-15- —.-.-........: C T2=6.2 8*A* A* DX*ROW*C PW*OEL TA/AOVER B /2.0/CONSTI 0316 DO 59 I=l,M 0317 C='T( I.,N-+1 -+T(.I +I N+I ) 0318 DQS=DQS+(C-C4 ( I,N+1 } ) )*C'2 0319 59 - C4( IN+I)=C 0320 00 60 J=lK2 0321 60 DR1(J)=3.14*ROL*CP*(A** 3)*DX*ER2*(2.0*J-1.0)/AOVERB 0322 DO 61 =1,tM 0323 DO 61 J=1,N 0324 C = (T( I,J)+T(IJ+1)+T(I+I,J)+T( I+,J+1) )/4.0/CONST1 0325 DQL = DQL -+(C-C4 ( I,J) )*DRI (J) 0326 61 C4(1tJ) = C 0327 DQI =(628*A*A*X*QSURFL/AOV ERB+DQ ILOT )*DT*CONST 0328 DMB = ( DQI-DQL-DQS)/HFG 0329 IF(DMB.LT.O.) OMB=-O. C330 DMBfT=DMBT+DMB 0331 DMTT=DMT-DMBT 0332. UU=UU-AOVERB*DMB/3.14/( A**3 ) /ROL/D'T 0333 DWDT=-. 581*PRESS*ALPHA*A*UU/AOVERB 0334 UG= —UU*( R/RO(L M.-+ t 1,)-I.. 0) 0335 E i= i-Q Z'5-DY 5 0336 E2=1-'DZ5-DYI 0337 E3=1 —2*OZ7-2*0 Y7 0338. DO 62 I[=K4 P 033'9 DO 62'J =1i'N 0.340 C 1 ( IJ )=DZl1U( I,J) 86

FORTRAN,IV G COMPILER MAIN 12-12-68 18:56.22 PAGE 0008 0341 62 C3("ItJ)=V(I*J) *ER4 0342 T( P+1,1 )=TO(P+1 t 1)*El+TO(P )*DZ5+TO(P+1,2)*DY5 1 +M4*M6*(M5+TO(P+ltl,)) 0343 - DO 63 J=2,N 0344 63 T(P+1 J)=TO(P+1,JI*E2+TO(P,J)*DZ5+TO(P+1,J-1 )*DRS5(J)+TO(P+1,J 1 +1)*0R6(J)+M4*M6*(MS+TO(P+ 1,J) 0345 DO 65 I=K4,P 0346 IF(U(I,1).LE.0.) GO TO 64 0347 C=(TO(I-l,1)-TO( I1))*C1(I,1) 0348 GO TO 65 0349 64 C=(TO(I~1)-TO( I+1,1))*Cl(l1) 0350......... 6'5.........T(I 1 )=T0(I l.:)*Ei+TO(1I,2)*DY5+(TO ( 1+i; 1 1)+TO( I-1 1) )*DL3 +C 1 +M4*M6*(M5+TO(I,1)) 0351 DO 68 I=K4,P 0352 DO 68 J=2,tN 0353 IF(U(I J).'LE.0.) GO TO 66 0354 C=(TO( I-1,J)-TO( IJ )) *Cl (,J) 0355 GO TO 308 0356 66 C=(TO( I J)-TO( 1 +LJ) ) *Cl (I,J ).035"7"". 308 IF(V(IJ).LE.O.) GO TO 67..... 0358 fD=(TO( IJ-1)-TtO(IJ))*C3(IJ) 0359 GO.TO 68 0360 67 D=(TO( I,J)-TO(IJ+i))*C3(1,J) 0361. 68 T(I.J)= TO ( I, j )*E2+(TO( I+1, J)+TO(I-1,J) )*DZ3+ 1 TO(1IJ-1)*DR5(J)+TO(I,J+1)*DR6(J)+C+D +M4*M6*(M5+TO(IJ)) 0362 DO 71 I=K4,P 0363 00'71 J=2,N 0364 FI=A4(J)*(-'(2*(U(1,J)-UO(IJ) )+CL(IJ)*(U( [l lJ)-U(I-lJ))4 1 C3(I,J)*(U(I,J+1)-U(I,J-1)J)*(RO(I,J+t)-RO(IJ-1))+AOVRB2*(2.....2.*(V(I,J)-V'O( I,J))+Cl (IJ) *(V(I+ 1,J)-V(I - 1J) ) +C3( IJ)*(V (IJ+i1 3 )-V(IJ-1)))*(RO(I+1,J)-RO(-1,J)))/RO(IJ)-W(I,J)*(2*(T(IJ) 4 -TO(I,J)W+C1( I,J-)*(T( I+1,J)-T(I-1,J) +C3UI,J)*(T([,J+1)-T(I,J 5 -1 ) )) )/ZOVR2/(T(I,J)+M5)+A6*W(I,J) 0365 I F(U( IJ) LE.O.) GO TO169 0366 C=(WO( I-,J)-WO( I,J)) *C( l tJ) 0367 GO TO 310 0368 69 C=(WO( I,J)-WO( 1+1,J))*C-1(IJ}) 0369 310 IF(V(IJ)A.LE.O.) GO 10 70 0370 D=(WOI(,J-l)-WO(IlJ))*C3(I,J) 037'1 GO TO 71 0372 70 D=(WO( I,J)-WO( I,J+l))*C3( I,J )..0373.....71....W(I,J)=WO(IJ)*E3+(WO(I+1,J)+WO(I-1,J))*DZ7 +C+iL) + I WO(It,J-1)*DR7(J)+WO(I,J+l)*L)R8(J)+FI 2 +A5*A4(J)*(T(IJ+1)-T(IJ-1))/CT(iJ)+M5) 0374 D00 72 J=2,N 0375...:SF('M+ 1 j)=RO( M+1,J )*UG*ER2*(J-10, )*( J-1.0)/2.0 0376'72 SFI(J)=SF(M+1,J). 0377 DO 73 I=K4,P 0378 73 SF( I,N+lI)=SF(M+IN+1l 0379 NE2=0 0380 74' NE2:NE2+1 0381 0382 00 75 I=K1tK3 0383 00'5J=1,K2 0384 75 C3( I-vJ)=SF(I,J) 87

FORTRAN IV G COMPILER _MAIN 12-12-68 18:56.22 PAGE 0009 0385 I=M+1 0386 76 1=I+1 0387 00 77 J=2,N 0388 - DI( J)= —(C3 I+1,J)+C3(I L,J )*F3-(W(I,J *RC( IJ)+A4(J)*(U(I,J)* 1 (RO( I J+ )-RO(I,J-1 ) )/D2R-V( I.JI*(RO( I+lJ)-RO( I-1J ))/D2)) 2 A1(J) 0389 77 F(J)=( DJ)}+F(J-1)*C2(J))/D2(J).0390 DO 78 J=2, A 0391 K=N+2-J 0392 78 SF(IK)=EJ1(K)*SF(I,K+1)+F (K) 0393 IF(I LT.P) GO TO 76 0394 IF(NT. EQ..oAND.NE2.EQ.1) GO TO 74 0395 RMAXX=O. 0396 00'79 I=K4 PNR 0397 DO 79 J=2,N,NR 0398 RR3= ABS ((SF( I J -C3(,J) )/ (C3( J ) +1.OE-20) ) 0399 79 IF(RMAXX.LT.RR3) RMAXX=RR3.C-400 - - - IF(RMAXX.GT-'.EPSLON) GO TO 74 0401 00 80 I=2,M 0402 - ---- -.80. — -..W ( TI —N'-i- ) — T1-)'S-F-''S'-tFF'-(-,-N-')' S ( I, * N+1 ) ) / DYY 0403 -.00 81 I=K4,P 0404 81' W(I-,N+1 )= (8*SF (ItN)-SF( I N-I)-7*SF(IN+1 ))/DYY/RO( IN+I) 0405 DO 82 J=2,N 0406 W(P+1,J)=(-SF(P-1,J)-+8.*tF(PJ)-7.*SF(P+1,J) )*R5(J)/RO(P+1,J) 0407 82 W(1,J) =(-SF(3,J ) +8.0*SF(2,J) )*R4(J) 0408 103 DO 83 I=KiK3 0409 00 83 J=1,K2 0410 V. I O('l,J'})=V(IJ) 0411 83 UO(IJ)=U(I,J) 0412.-DO 84 I=2,M 0413 U( i N)=2.0*(3.*SF(I N)-6.O*SF (I N-1) +SF ( IN-2) )/R1(N) 0414 U(I,1)=SF( I 2 )/ER6 0415 84 U( I2)=2.0*(6.0*SF(I,3)-3.0*SF( I 2)-SF( I,4))/R1 (2) 0416' 00 85 I=Ki,P 0417 U(IN)=2.*(3*SF(IN)-6*SF(IN-1)+SF(IN-2) )/RI(N)/RO(ItN) 0418 U.tl U' i')'v=S F(,2 )/ER6/RO(I, 1) 0419 85 U(It 2)=2.*(6.*SF(I,3)-SF(I,4)-.3.*SF( I2))/R-I(2)/RO(I,2) 0420U 00 86 1=2M - 04-21 DO 86 J=3,K5 0422 86 U(IJ )=(SF( IJ-2)-8*SF(IJ-l)+8*SF([IJ+t)-SF(ItJ+2))/Rl(J) 0423 00 89 I=KLP 0424 00 89 J=3,KS5 0425 89 U( I tJ)=( SF ( IJ-2)-8*SF( I J- )+8:*SF( I J+1 ) —SF ( I J -2) )/R (iJ 1 /RO(IJ) 0426 00 90 J=2,N. 0427 V(M+2"J )=2.*(3.*SF (M+2,J )+.SF(M+4, J)-6.*SF(M+3,J)+2*SF(M+1IJ)): 1 /R6(J) /RO(.M-+2 J') 0428 V(P,J)=2**(6.*SF(P- 1 J) 3.*-SF(P,J.)-SF(P-2,J ) )/R6(J )/RO(P,~J) 0429 90 VG(J)=2.*{(SF(M+3,J)-8.*SF( M+2J)+7.*SF(M+1,J )/R6(J)/RO(IM*+J 0430 0 91 I=K6 K7 0431 00 91 J=2,N - - 0432 91 V( I,J)=(SF( I+2, J)-8*SF( I+1,J)+8*SF( I-1,J)-SF(I-2,J3 )/R6(J)/RO 0 1 ( IJ) — 04-33 00O 92 J=2,K2 88

FORTRAN IV. G COMPILER MAIN 12-12-68 18:56.22 PAGE 0010 0434 92 SF(M+lJ)=O. 0435 DO 93 J=2-N 0436 V( 2J)=2'0O*(SF(4,J)-6.O*SF(3,J)+3.0*SF(2,J) )/R2(J) 0437 VIM, J)=2.0*(6.0*SF(M-,J)-3. O*SF(M-J)-SF(IM-2,J))/R2(J). 0438 93 V(M+1,J)=2.O*(8*SF(MJ)-SF(M-1~J))/R2(J) 0439 1 094 I=3,K8 0440.00 94 J=2,tN 0441 94 V( I J)=(SF ( I+2,J)-8*SF( 11,J)+8*SF(I-,J )-sSF (I-2tJ))/R2(J) 0442 IF(NT.LT.N1) GO TO 104 0443 IF(NT.NE.NMAX) GO TO 95 0444. PUNCH 204, APRE SS, PO. X 4.NT.T IME tUUODM TT DMTT, ( (T ( I,J),J=1 K2 ), 1I=1,K3)',((SF(IJ),tJ=lK2),I= 1,K3) 0445 GO T0.104 0446 95 NP1=TAU/(TAUL*N3) 0447 IF(NP1.GE'1..) GO TO 104 0448 GO TO 102 0449 104 IF(NT,'LT.NC) GO TO 105 0450 RRL=O 0451 DO 96 I=2,M 0452 00 96 J=2,K2 0453 RR2= ABS ((T(I,J)-TO(IJ))/(TO(ItJ)+1.E-201) 0454 96 IF(RRL.LT.R2) RR=RR2 0455 105 NUG=O. 0456 NUL=OC. 0457 RAL = 0. 0458 RAG = 0. 0459 DO 97 1=1,1(1 0460 HL(I)' ABS (KL*OTDRW(I)/A/(T(IN+1)-TCIl))) 0461 RAt = RAL+DX*TItIN-+I) 0462 9- 7 NUL = NUL+DX*HL()*A*.X/AOVERBO/KL 0463 RAL = RAL*(X**~ 3)/(AOVERB** 4) 0464 13 98 I=K1,K3 0465 J=I-M....046'6..............HG(J) ='ABS (KG*OTDRW(I)/A/(T( I,N+1)-T(I,1) )) 0467 RAG = RAG+DZ*"T(I,N+I) 0468 98 NUG = NUG+0Z*HG(J)*A*( l-X)/AOVERB/KG 0469 RAG = RAG.* ((l-X)** 3) * BEITAG/RALPHA/RNEW/BEITA/ I (AOVERB** 4) 0470 CL = NUL/(RAL:** 0.25) 0471 CG = NUG/(RAG** 0.25)'. 0472 UR=O. -04 73 DO J=2,K 0474 99 UR=(UR+U( I,J )*R1 (J))/12.0 0475 DO 401 1=1,KK3 0476 o00 401 J=1,K2 047- 401 R T( I'.J )':CC:NST3*T(I,J) 04178 WRITE(6,210) NT,'TIME,'TAUDTURUUPRESStZtXt DXtDMEtDMTDMBDMBT, ID-M',MTT-,RQIN,' DQI'DT,'DWDT'TDQWDT,HFG T SA T I N T 0479 210 FORMAT(lHI,' NT= I113,' TIME=' E13.6, _______________-1 ___ - TTAU=' F13.6 ~' T'-'.-1-'36// 21X, _ UR=' E13.6, " UU='E13.6, 3 - PRESS=' f3 = F13.6// 41X,' X=' F13.6,' DX=' F13.6, _ _ _ _ _ _ _ _ _ ________- l9gE ='E 1 3., ODMT=' E 13.6/ / 61X"' DMB=" F13.6,' DMBT=" F.13.6, 89

FORTRAN IV G COMPILER MAIN 12-12-68 18:56.22 PAGE 0011 7' DM=' E13.6,' OMTT=' E13.6// 81X' RQIN=' E13.69' DQIOT='.E13.6, 9 DWOT=' F13.6, _ DQWDT=' F13.6// 1 1 X' HFG=' F13.6, TSAT=' F13.6 2' TINIT=' F13.6/11/ 320X,' AXIAL VELOCITY COMPONENT —U(-IJ)')' // ) 0480 00 601 I=1',K3 0481 601 WRITE(6,211) I,(U(I,J)oJ=1,K2) 0482 211 FORMAT(lHO,13,7('5XE13.6)/4Xt7(5XE13.6)/4X,7(SXE13.6)) 0483 WRITE(6,212) 0484 212 FORMAT(.//20X,'RADIAL VELOCITY COMPONENT —-V(I,J)'//) 0485 DO 302 1=1,K3 0486 302 WRI-TE(69211 ) I,(V(IJ),J=1,K2) 0487 WRITE(61213) 0488. 213 FORMAT( //20X,'NON DIMENSIONAL TEMPERATURE RISE —-T(I,J)'//) 0489 DO 603 1=1,K3 0490 603 WRITE(6,211) I,(T(IJ)vJ=1,K2) 0491 W-RiTEI-6,72'2V-2-...-_.. 0492 222 FORMAT(//20X,'REAL TEMPERATURE RISE —-RT(IJ)'//) 0493 DO 608 I=l.K3 0494 608 WRITE(6,211 ) I,(RT( I,J),J=1,K2) 0496 214 F]RMAT(//20Xt'STREAM FUNCTION —-SF(I,J)'//) 0497 DO'304 11,K3 0498 304 WRITE(6,211) I,(SF( IJ),dJ=1K2) 0499 WRITE(6,215) 0500 215 FORMAT(//20XeRADIAL VAPOR VELOCITY COMPONENT AT INTERFACE —IVG(J)'//) 0501 WRITE(6,216) (VG(J),J=lK2) 05 05'2 —- -"-....21f6...F —ORMAT(4X,7(5X,'EI3.6)/4X75X, (X E13.6)/4X,7 ( 5XE13.6)) 0503 WRITE(6,217) 0504 217 FORMAT(//20X,'[NTERFACE VELOCITY DUE'TO. IN'T-ERFACE CHANGE —-ULJ)' 0505 -...-... WRITE(6"'-'216) ('' UL(J), J= 1, K2) 0506 WRITE(6,218)...-0507 21. —..'.'8 2$.FORMAT(//2OX,.'VAPOR STREAM FUNCTION AT INTERFACE —-SFI(J)'//i) 0508 WRITE(6,216) (SFI(J)tJ=1,K2) 0509 WRITE(6,219) 0510 219 FORMAT(//20X,'HEAT TRANSFER COEFFICIENT OF LIQUID —-HL(J)'//) 0511. I TE ( 6e216 ) (-HL J, R=''........K1) 051.2 WRITE(6,220)...051-3 F.....T.220.F'OR'MAT(//20-X,'HEAT TRANSFER-COEFFICIENT OF VAPOR —-HG(J)'//) 0514 WRITE(6,223) (HG(J),J=1,11) 0515 223 FORMAT (4X,7 ( 5X, E13.6 )/4.X,7 ( XE13.6 ) ) 0516 WRITE(6,221) NUL,NUGRMAXRRI,NE1,NE2,RAL,RAGCLCG, RP, DQILDT, IDQIvOQSIDQL 0517 221 FORMATI1HO,' NUL=' F13.6': NUG=' F13.6, ~~~~~- 1 ~ ~' RMAX= E13".6, 6 RR'=''E13.6// 21X;"' NEI=' 113, NE2=' 113, 3, —' RAL=' E13.6.,'RAG=' E.3';-"6/'/'....... 41X,' CL=' F13.6,'.. CG=IFL3.6, 5 R P ——' —'F36,' D-Q i-LDT=-'f].'-3'"-'/ F3. 61X,' - DQI=' E13.6, " DQS= E13.6, 7 7 *DQL' E13.611T) -_ _0518 N3=N3+1 90

FORTRAN IV G COMPILER MAIN 12-12-68 18:56.22 PAGE 0012 0519 NP2=TAU/ (N4*TAU2 ) 0520 IF(NP2.GE.1. ) N4=N4+1 052-1 IF(NP2.GE.-1.) WRITEI7,204'APPRESSPOtX,NTTIMEUUtDMBT,DMTT, I( MTI 9J).J=l-VK2),I=l. K3) t ( SF( I- J).J=1 tK2),"Il-. tK3). —----- 0522 -IF(NE1.GT.NEI GO TO 500 0523 IF(NT.EQ.NMAX)- GO TO 101 0524 GO TO 102 0525 500 CALL SYSTEM -0526 END 91

APPENDIX C LH2 PROGRAM NOMENCLATURE 93

Symbols preceding the expression indicate the following: - Input data ** - Printout Symbols following the expression indicate the following: [0] - Integer [1] - Dimensionless [1 - Units () - Statement number in the source program of Appendix B A *A - [ft] - Tank radius *AOV1RB - [1] - A/B AOVERB2 - [1] - (A/B)2 - (46) A5 - [1] - (250) A6 - [1] - (243) Al( J) - [1] - (48) A2(J) - [1] - (93) A3(J) - [1] - (94) A4(J) - [1] - (50) *ALPHA - [ft2/sec] - Thermal diffusivity of liquid I*ALPHAW - [ft2/sec- — Thermal diffusivity of wall *ALPHAV - [ft2/sec] - Thermal diffusivity of vapor B BB - (30,32,246) - Temporary storage B - [ft] - (19) - Total tank height BJ - [1] - (88) BJl - [1] - (81) BO - [1] - (89,148,151) BOO - [1] - (147,150) *BEITA - [F- 1] - - Volumetric coefficient of expansionliquid BEITAG - [~F-1] - (68) - Volumetric coefficient of expansionvapor *Bl - (30,246) - Pressure temperature relation *B2 - (31,247) - Pressure temperature relation *B3 - (30,246) - Pressure temperature relation *B4 - (30,246) - Pressure temperature relation C C - [1] - Temporary storage Cl(l, J) - [1] - (202,340) - Temporary storage C2(J) - [1] - (49) C3(IJ) - [1] - (203,341,384) - Temporary storage C4(I-r-J) - [1] - (319,326) - Temporary storage **CL - [1] - (470) - Coefficient of Rayleigh- number correlation-liquid **CG - [1] - (471) - Coefficient of Rayleigh number correlation-vapor ACP r- [Btu/lbm — F] - Specific heat-li'quid *CPV - [Btu/lbm-~F] - - Specific heat-vapor-constant pressure *CV - [Btu/lbm-~F] - - Specific heat-vapor-constant volume *CPW - [Btu/lbm-~F] - Specific heat-wall CT - [1] - (70) - Property term for vapor (=1 for ideal gas) 94

CT2 - [Btu] - (315) - Property term for wall *CODE - [O] - (52,103,139) - = 2 for a new run = 1 for continuation of prior run via READ FORMAT INPUT **CONST - [sec] - (35) - Conversion constant for dimensionless time **CONST1 - [~F-l] - (36) - Conversion constant for dimensionless temperature **CONST2 - [sec- ] - (37) - = 1/CONST **CONST3 - [~F] - (38) - = 1/CONST1 D D - [Ill - Temporary storage D1(J) - [1] - (95) D2(J) - [1] - (97) DI(J) - [1]' - (388) DR1(J) - [1] - (191,321) DR2(.J) - [1] - (192) DR3(J) - [1] - (193) DR4(J) - [1] - (194) DR5(J) - [1] - (196) DR6(J) - [1] - (197) DR7(J) - [1] - (198) DR8(J) - [1] - (199) DZX - [1] - (74) DZX1 - [1] - (75) DZX2 - [1] - (76) DZXH - [1] - (77) DR - [1] - (24) - Radial grid space D2R - [1] - (45) - DR/2 DRZ - [1] - (79) - (DR/DZ)2 **DX - [1] - (71) - Axial grid space-liquid DX1 - [1] - (163) - DT/DX DX2 - [1] - (72) - DX2 DX3 - [1] - (176) DX5 - [1] - (180) DX7 - [1] - (182) DX8 - [1] - (183) DXR - [1] - (82) - (DX/DR)2 DRX - [1] - (83) - 1/DXR DYY - [1] - (27) - 2(DR)2 DY1 - [1] - (173) DY3 - [1] - (172) DY5 - [1] - (175) DY7 - [1] - (174) DZ - [1] - (73) - Axial grid space-vapor DZ1 - [1] - (164) - DT/DZ DZ2 - [1] - (78) - DZ2 D2Z - [1] - (86) - DZ/2 DZ3 - [1] - (165) DZ5 - [1] - (166) DZ7 - [1] - (167) Dz8 - [1] - (168) DZ88 - [1] - (169) **DM - [lbm] - (21I) - Net phase change -in time DT (minus sign for evaporation) **DMB - [lbm] - (314,328) - Phase change in time DT from energy balance on tank wall and liquid system (plus sign for evaporation)

**DME - [lbm] - (239) - Phase change at L-V interface in time DT (minus sign for evaporation) **DMT - [lbm] - (240) - Cumulative total of DME (minus sign for evaporation) **-DMBT - [lbm] - (330) - Cumulative total of DMB **DMTT -' [lbm] - (331) - Cumulative total of DM (minus sign for +* [1] -evaporation) **DT - [1] - Time step interval *DTS - [~F] - (311) - Max. permissible liquid superheat at node adjacent to wall *DTW - [~F] - (310) - Max. permissible superheat of wall in contact with liquid DTDRW(I) - [1] - (226,229) - Temperature gradient in fluid at wall DTDXG(J) - [1] - (220) - Temperature gradient in vapor at liquidvapor interface DTDXL(J) - [1] - (219) - Temperature gradient in liquid at liquidvapor interface DQIL - [1] - (231,233) - Mean temperature gradient in liquid at L-V interface DQILDT - [Btu/sec] - (237) - Heat transfer rate in liquid at L-V interface (plus for heat transfer to liquid) **DQIDT - [Btu/sec] - (238) - Heat transfer rate in vapor at L-V interface (plus for heat transfer out of vapor) **DQWDT - [Btu/sec] - (235) - Heat transfer rate to vapor from wall (plus for heat transfer to vapor) **DWDT - [Btu/sec] - (333) - Work rate by vapor control volume DQI - [Btu] - (327) - Sum of heat transfer to wall in contact with liquid and to liquid at L-V interface in DT (plus for heat transfer in) DQL - [Btu] - (313,325) - Enthalpy rise of liquid in DT DQS - [Btu] - (312,318) - Enthalpy rise of wall in contact with liquid in DT *DELTA - [ft] - - Wall thickness E EJ(J) - [1] - (90,102) EJ1(J) - [1] - (91,98) *EPSLON - [1] - (308,400) - Maximum fractional change in stream function between iterations before iteration.is terminated El - [1] - (185) E2 - [1] - (186) E3 - [1] - (187) EI - [1] - (190) ER1 - [1] - (181) ER2 - [1] - (25) ER3 - [1] - (177) ER4 - [1] - (178) ER5 - [1] - (179) ER6 - [1] - (28) ER7 - [1] - (184) F - - - Temporary storage F(J) - [1] - (295,298,389) - Temporary storage FI - [1] - (189,364) - Temporary storage 96

F1 - [1] - (84) F2 - [1] - (85) F3 - [1] - (80) *FHHFG - [Joule/gram mole] - (62) - Input function value of HFG in the subroutine ATSM G *G - [ft/sec2] - - Acceleration corresponding to effective body force acting on container GAM - [1] - (26) - Ratio of specific heats-vapor H H - [0] - (300) - Index **HFG - [Btu/lbm] - (64) - Latent heat of vaporization corresponding to TSAT HGFJ - [Joule/gram mole] - (63,64) - HFG in Joule/gram mole corresponding to TSAT obtained by subroutine ATSM **HG(I) - [Btu/sec-ft2- F] - (466) - Local heat transfer coefficient of vapor based on AT between wall and centerline **HL(I) - [Btu/sec-ft -~ F] - (460) - Local heat transfer coefficient of' liquid based on AT between wall and centerline HHFG - [Joule/gram mole] - (62,63) - Function value of HFG in given tableHFG vs. TSAT appearing subroutine ALI and ATSM I I - [0] - - Axial nodal index number Il - [0] - (17) - Mid-vertical height of liquid IER - [0] - (63) - Error parameter for subroutine ALI J J - [0] - Radial nodal index number K *KG - [Btu/sec-ft-~F] - - Thermal conductivity-vapor *KL - [Btu/sec-ft-~F] - - Thermal conductivity-liquid K1 - - (8) - (M + 1) K2 - - (9) - (N + 1) K3 - (10) - (P + 1) K4 - (11) - (M + 2) K5 - (12) - (N- 1) K6 - (13) - (M + 3) K7 - - (14) -(P- 1) K8 - - (15) - (M- 1) M *M - [0] - Number of vertical divisions in liquid (M + 1 = L-V interface) M1 - [1] - (40) - Liquid-wall property M2 - [1] - (41) - Liquid-wall property M3 - [1] - (42) - Vapor-wall property M4 - [1] - (69) - Vapor property M5 - [1] - (44) - Initial temperature M6 - [1] - (244) - Pressure rise ratio in DT M7. - [1] - (43) - Vapor-wall property Ml - [1] - (170) M23 - [1] - (171) 97

N *N - [0] - Number of radial divisions (N + 1 = wall) *Nl - [1] - (442) - Minimum number of time steps before punch is permitted N3 - [0] - (57,518) - Control number for establishing time steps between printouts N4 - [0] - (58,520) - Control number for establishing time steps between punchouts *NC - [0] - (449) - Minimum number of time steps NT before fractional temperature changes, between time steps is computed *NE - [0] - (381) - Maximum number of iterations permitted in computations of stream function **NE1 - [0] - (287,288,289) - Counter on iterations on stream functionliquid **NE2 - [0] - (379,380,381) - Counter on iterations on stream functionvapor NP1 - [1] - (446,447) - Control for printout instruction NP2 - [1] - (519) *NR - [0] - (396,397) - Index steps of grid spaces for which fractional change in stream function with iterations is computed **NT - [0] - (60,523) - Time step number *NMAX - - (523) - Maximum number of time steps NT permitted before program is terminated *NEW -.[ft2/sec] - - Kinematic viscosity-liquid *NEWV - [ft2/sec] - - Kinematic viscosity-vapor **NUG - [1] - (455,468) - Mean Nusselt number-vapor **NUL - [1] - (456,462) - Mean Nusselt number-liquid *NDIM - [0] - (62,63) - Number of points which must be selected out of given table (ZTT,FHHFG) P *P - [0] - - Total number of vertical divisions *PO - [lbf/in.2] - - Initial system pressure P1 - [lbf/in.2'] - (59,245) - Pressure at previous time step **PRESS - [lbf/in.2] - (29,242) - Current system pressure **PR - [1] - (20) - Prandtl number-liquid **PRV - [1] - (21) - Prandtl number-vapor QI - [1] - (230,234) - Mean temperature gradient in vapor at L-V interface QW - [1] - (227,229) - Mean temperature gradient in vapor at wall *QSURFG - [Btu/sec-ft2] - - Imposed heat flux on container exterior on vapor portion *QSURFL - [Btu/sec-ft2] - - Imposed heat flux on container exterior on liquid portion R **RR1 - [1] - (450,454) - Maximum value of R2 RR2 - [1] - (453,454) - Fractional change in temperature between time steps RR3 - [1] - (306,307,398, - Fractional change in stream function with 399) iteration R1(J) - [1] - (51) R2(J) - [1] - (96) 98

R3(J) - [1] - (195) R4(J) - [1] - (99) R5(J) - [1] - (100) R6(J) - [1] - (101) RO(I,J) - [lbm/ft3] - (113,217) - Local vapor density *ROL - [lbm/ft3] - - Liquid density *ROW - [lbm/ft3] - - Wall density ~**RP - [1] - (65) - Ratio of current pressure to initial **RMAX - [1] - (303,307) - Maximum value of change in stream function with iteration-liquid RMAXX - [1] - (395,399,400) - Maximum value of change in stream function with iteration-vapor **RAG - [1] - (458,467,469) - Mean Rayleigh number-vapor **RAL - [1] - (457,461,463) - Mean Rayleigh number-liquid *RGAS - [ft-lbf/lbm-~R] - - Gas constant-vapor **RQIN - [1] - (236) - Fraction of total heat transfer rate to vapor part of tank going into vapor RALPHA - [1] - (23) - Ratio-thermal diffusivity of vapor to liquid RNEW - [1] - (22) - Ratio-kinematic viscosity of vapor to liquid RT(I,J) - [~F or ~R] - (477) - Real temperature rise S S - [1- (144,154,155 - Stability criteria 157,158) SMAX - [1] - (155,158) - Maximum value of S **SF(I,J) - [1] - Stream function **SFI(J) - [1] - (376) - Vapor stream function at L-V interface T **T(I,J) - [1] - Current temperature TO(I,J) - [1] - (56) - Temperature of prior time step **TAU - [sec] - (161) - Current time *TAU1 - [sec] - (446) - Basic multiple of time for which printout occur s *TAU2 - [sec] - (519) - Basic multiple of time for which punchout occurs **TIME - [1] - (160) - Current time TINIT - [~R] - (34) - Initial saturation temperature corresponding to initial pressure TSAT - [~R] - (33,249) - Current saturation temperature corresponding to current pressure TSATK - [~K] - (61,62,63) - Saturation temperature in OK in subroutine ALI and ASTM TT - [~K] - (62,63) - Argument value for temperature in the subroutine ALI U **U(I,J) - [1] - - Axial component of velocity **UU - [1] - (224,332) - Mean velocity of liquid-vapor interface UG - [1] - (116,334) - Mean velocity of vapor at liquid-vapor interface **UL(J) - [1] - (221) - Local L-V interface velocity due to interface phase change UO(I,J) - [1] - (411) - Temporary storage of U(I,J) **UR - [1]. - (472,474) - Mean axial velocity at mid-point of liquid (serves as check on continuity) 99

V 8 **V(T-J) - [1] - Radial component of velocity VOkI,J) - [1] - (410) - Temporary storage of V(I,J) -*VG(J) - [1] - (128,429) - Radial velocity of vapor at L-V at interface w W(I,J) - [1] - Vorticity WO(I,J) - [1] - (213) - Vorticity of prior time step X C***X - [1] - (162) - ~iquid fraction in container Y - [1] - (18) -'A/B Z **Z - [1] - Compressibility factor, Z = 1 for ideal gas ZOVR2 - [1] - (87) - = Z/2, = 0.5 for ideal gas *ZTT - [~K] - (62) - Input argument value of temperature in the subroutine ATSM 100

APPENDIX D LH2 PROGRAM —DATA INPUTS 101

CODE [0] = 2 for a new run = 1 for continuation of prior run via format input DT [1] Initial estimate of dimensionless time step EPSLON [1] Maximum fractional change in stream function between iterations before iteration is terminated M [0] Number of vertical grid spaces in liquid N [0] Number of radial grid spaces NC [0] Minimum number of time steps NT before fractional temperature changes between time steps is computed NDIM [0] Number of points which must be selected out of given table (ZTT,FHHFG) NE [0] Maximum number of iterations permitted in computations of stream function NMAX [0] Maximum number of time steps NT permitted before program is terminated NR [0] Multiple of index steps of grid spaces for which fractional change in stream function with iterations is computed N1 [0] Minimum number of time steps NT before punchout format is executed P [lbf/in.2] Total number of vertical grid spaces TAU1 [sec] Basic multiple of real time for which printout is executed TAU2 [sec] Basic multiple of real time for which punchout is executed A [ft] Tank radius ALPHA [ft2/sec] Thermal diffusivity of liquid 102

ALPHAV [ft2/sec] Thermal diffusivity of vapor ALPHAW [ft2/sec] Thermal diffusivity of tank wall AOVERB [1] Ratio of tank radius to height BEITA [~F-1] Volumetric coefficient of expansion of liquid B1,B2,B3, and B4 Coefficients in pressure and temperature relation CP [Btu/lbm-~F] Specific heat of liquid CPV [Btu/lbm-~F] Specific heat of vapor, constant pressure CPW [Btu/lbm-~F] Specific heat of tank wall CV [Btu/lbm-~F] Specific heat of vapor, constant volume DELTA [ft] Tank wall thickness DTS [~R] Imposed maximum permissible temperature difference between liquid nodes adjacent to tank wall and saturation DTW [OR] Imposed maximum permissible temperature difference between tank wall and saturation FHHFG (Joule/gram mole] Input function values of heat of vaporization in table between temperature and heat of vaporization G [ft/sec2] Acceleration corresponding to effective body force acting on tank KG [Btu/sec-ft-~F] Thermal conductivity of vapor KL [Btu/sec-ft-~F] Thermal conductivity of liquid NEW [ft2/sec] Kinematic viscosity of liquid NEWV [ft2/sec] Kinematic viscosity of vapor P0 [lbf/in.2] Initial system pressure 103

QSURFG [Btu/sec-ft2] Imposed heat flux on tank exterior on vapor portion QSURFL [Btu/sec-ft2] Imposed heat flux on tank exterior on liquid portion RGAS [lbf-ft/lbm- ~R] Gas constant of vapor ROL [lbm/ft3] Liquid density ROW [lbm/ft3] Density of tank wall X [1] Initial fraction of liquid in tank Z [1] Compressibility factor ZTT [OK] Input argument values of temperature in table between temperature and heat of vaporization 104

APPENDIX E LH2 PROGRAM-TYPICAL OUTPUT (RUN B-H 47) 105

$RUN CYLIGH+*SSP. 5=*SOt-RCE* 6=*SINK* 7=*PUNCH* EXECUTION BEG!NS &DATA]1 A= 11.000000,AOVER8-= 0.376999S7 _= 30t*M= 20 N= 20.tK- 0.32299995 _* CODE- 2 DT= 0.99999943E-06,NMAX= 1644,PO= 12.400000,G= O.55000000E-02,QSURFG= 0.43199994E-02,QSURFL= 0.5999996E-01,DTW 1.0000000,DTS= 1.000000Q _,ALPHA=. 0O16599997E-O05NEW= 0.20649995E-05.KL= 0.18899998E-04,8ETA- O.86199977E-O2,ROL= 4.3899994 #CP- 2.5999994,ALPHAV= 0.10600000E-04,N.EWV= O84999992E-05,KG= O.24999999E-O5,CPV= 2.8599997',CV 1.4919996 RGAS= 766.39990.ALPHAW= 0.1-9849995E-05,CPi= 0.34999996 VROW=.10.000000 ELTA= 0.59099998E-O1,TAU1=, 2.00OO,N1 1NE= 160,EPSLCN= 0.99999979E-02,NC= 5,NR= 4,81= 3.1679363,B2= -90O.174744 B83= 1.8079195 __,__.80 S?.138.86E-02,Z-.O_1 OCCOOOO.~._.ZT.T= 18.000000 _, I 19.. 000000.20000000_ 21.000000, 22.00000, 23.000 24.000000 25.000000 26.000000 t 27.000000, 28.000000 29.000000 f~HHFG= 911.59985.2980_ 900.19995, 898*29980 890.59985 _ 860.50000, 839.69995 - 814.59985, 784.59585, 704.29980, 651*29980.NDIM= 5#TAU2 1500 &END, _' &DATA2 CONST= 72891568..CCNST2= L0.13719006E-07,C 0NST1 —N.I=3 —. 0 C = 4409178E-09,PR7 1. &END O

NTh 1644 TIME'= 0.646461E-04 TAU= 4712.152344 DT= 0*287711E-07UR= 0.168978E 04 UU=-C.414617E 02 PRE5rZt- -18.574661 Z= 1.000000 X=, 0.32C767: DX= 0.016038 0tME=-0*234399E-0-i DMI=-0.439927E 00.Dms8 0.057820 Ot4BT= 106.511505 DM=-0.568196E-01 DMTT=-0.106951E.03 RQIN — C.120051E C0 DQIDT= 0.395722E-02 DWD.T= 0.021672 OQWDT= 0.710023 HEG=- 193.040C-54 TSAT= 37.956543 TINIT= 35.468872___ AXIAL VELOCITY CUM PONEN T.-' —U(It J) 1, 0.0' 0.0 0.0 0.0. 0.0 0O.0 0.0O 0.0 0.0 __ 0.0 __0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0. 0.0 2 -0*378249E 05 -0.380702E C5 -0.391893E 05 -0o404994E 05 -0.419683E 05 -0.433900E 05 0450E5 -0*450434E 05 -0.446796E 05 -0.430679E 05 -0.397489E 05 -0.339494E 05 ~0. 234420E- 05 0.1247E0 0.249.151E 05 0.320525E C5 0.299789E 05 0.178306E 05 -0.258490E 05 0.920880E 04 0.0 3 -0.742263E'05 -0.746481E 05 -0.767244E OS -0.791901E 05 -0.819737E 05 -0.846552E 0.5 -0.629E0'-0.876438E 05 -0.868214E 05 -0.836477E 05 -0.7-74085E 05 -0.671303E 05 -0*512657E 05 -0*2780E0 -00137634E 04 0.2I?88E 05 0.354701LE 05 0*351857E 05 0.2.56945E 04 0.833868E 05 0.0 04 -0*108251E 06.-0.108802E 06 -0.111703E-06 -0.115181E 06 -0.119112E 06 ~-0.122863E 06 -0.1259E0 — 4 ~ —Q0 126825E 06 -0.1259466E 06 -0.120-843E 06 -0.112164E 06 -0.985822E 05.-0*792690E 05'-0.54331!0 -0.251982E 05.0.418896E 04 04310598E 05 0.497999E 05 0.380904E O 0*160392E 06 0.0 5 -00139174E 06 ~-0.139825E 06 -0.143438E 06 -0.147793E 06 -0.l152700E 06 -0. 157329E 06'-0.13E0 -0.16L927E 06 -0.159983E 06.-0.154029E 06 -0*143267E- 06 -0.126975E 06 ~-0.104651E 06 -O.:6462I0 -0.436.535E 05 -0*802578E 04 0.286538E 05 0.629300E 05 0.681690E 05 0.226372E 06 0.0 6 -0*166405E 06 -0.167137E 06.-O.171362E 06 -0.176464E. 06 -0.182183E 06 -0.187509E 06 -0.1911E0 -0.,192470E 06 -0.189926E C6 -0.182759E 06 -0.170171E 06 -0.1I51485E 06.-0.126261E' 06 -00.9588E0 -0.572845E 05 -~0.154527E 05 1).296418E 05'O.7.54031E 05 0.907488E 05 G.2785'79E 06 0.07. -0.1-89459E 06.-O. 1902 59E C6 -0.195001E 06 -0*200727E 06 -0.207106E- 06 -0o212978E 06 -0O1765 0 -0.218114E 06 -0.214996E 06 -0.206752E 06 -0.192565E 06 -0* 171725E 06 -0.143716E 06 -001088E0 -0*666256E 05 -0.189586E 05 -0.334080E 05 0.'876415E 05 0 106495E 06 0.316748E 06 0.0.8 -0.0207944E 06 -0.208601E 06 -0.213'971E 06 -0.:220204E 06 -0.227111E 06. -O0.233407E'06 0269E6 -0.0238625E 06 -O0.235010E 06.-0.225859E 06 -0.'210322E 06 -0. 187602E 06 -0*157025E 06 -0.1187E0 -0.722409E 05.-0.193060E%05 0.395220E 05 0.101129E.06 0.119459E 06 0*341486E' 06 0.0 9 -0.221545E 06 -0*'222450E 06 ~-.227961E 06 ~-0.234596E 06 -0.241919E 0.6 ~0*248548E 06 -O0.25997 0 -0*253844E: 06 0.-O249859E 06 -0.240025-E 06 -0.223439E 06 -0o'L99191E 06 -0*166413E 06 -0.1241E0 -0.0747902E 05- -0.173435E 05 0*472957E 05 0.115206E 06 0.1L6553 7E 06 0*373586E.06 0.0 10 -0*2-30026E.06 -0 230968E 06 -00236737E 06 -0.243676E 06 -0.251318E 06 -~0*2582L8E 06 -0.2624 0 -0.0263668E 06'-00259487E 06 -0.249241E 06 -0.231977E 06 -0.206663E- 06 -0.172231E 06 -0.1Z89E0 -0*.750733E 05 -0.141911E 05 0.551203E.05 0.12861-5E 06 0.210632E- 06 0*396591E 06 0.0 II -0.233216E.06 -0. 234184E: C6 -0024012SE: 06 -0.247277E 06' -00.255155E 06. -0*.262280E 06 -0.2607E0 -00'268019E 06.-0.26385-3E 06 -00.253514,E 06 -0*2360I0E:06 -0.210212E 06 -04174879E 06 -0.1Z93E0 -00'740239E. 05 -0.*110449E 05 0.611046E 05' 0.138815E 06 0.241424E 06 0*406940E 06 0.0

12 -0.231018E 06 -O.231999E 06 -0.238031E 06 -0.245295E 06 -0.253326E5 06;-0.260639E_06 -0.2654E0 -0.266830E 06 -0.262913E 06 -0.252830E 06 -0.235576E 06 -0.209981E 06 -0.174701E 06 -00.128 E0 -0.727905E 05 -O.87-9692E 04 0.623E0.438E0.616 6 0406961E 06 0.0 13 -0.223407E 06 -0.224382E 06 _-0.230407E 06 -0.237683E 06 -0.Z45776E 06 -0.253233E 06 -0.2588E0 ~-0.260037E 06.-O.256600E 06 - -0.247125E 06 -0.230634E 06 -0.205998E 06 -0.171870E 06 -O.1275E0 -0.719883E 05 -0.804719E C4_ 0.650273E 0.5 0.148934E 06 0.270015E 06 0.398148E 06 0.0 14 -0.2 10444E 06 -0.211389E 06 -0.217295E 06 -0.224459E 06 -0.232501E 06 -0.240035E 06 -0.2454E0 -0.247561E 06 -0.244816E 06- -0.236275E C6 -0.221029E 06 -0, 198064E 06- -0.166135E 06 -0.1243E0 -0.713168E 05 -0.867249E C4 0.639010E_05 __ 0.149782E 06 0.271190E 06 0.380439E 06 0.0 15 -0.192282E 06 -0.193172E 06 -0.198824E 06 -0.205721E 06 -0.213556E 06 -0*221048E 06 -0.2267k0 -0.229314E 6 -027 06 0.22017 06 -0*206506E 06 -0.185769E 06 -0.156835E 06 -0.1187E0 -0*696769E 05 -0.995604E C4 0.609483E 05 0.146487E 06 0.262142E 06 0.353475E 06 0.0 16 -0.169193E 06 -0.169999E 06 -0.175233E 06 -0.181666E 06 -0.189079E 06 -0.196339E 06 -'0.0255E0 -0.205210E 06 -O.2043OOE- 066 -0.198428E 06 -0.186800E 06 -.1864E0 -0.143266E 06 -0.1090E0 -0.657874E 05 -0.110870E CS 0,555987E 05 0.136862E 06 0: 241063E 06 0.317529E 06 0. _____ 11 -0.141586E 06 -0.142279E 06 -0.146895E 06 -0.152620E 06 -0.l59323E 06 -0*166065E 06 -0.1714E0 -0.175183E 06 -0.1715276E 06 -0.171046E 06 -0.161711E 06 -0. 146619E 06 -0.125110E 06 -0.6373 0 -O.590221E 05 -0.116766E 05 _ 0.471807E 05 __.19321E 06 0.208787E 06 0.273584E'06 0.0 __ 18 -Q.110O28E 06 -0.110579E 06 -0.114355E 06 -0.119080E 06 -0.l24710E_06 -0.130533E- 06 -.0.1358 0 -0.l39228E 06 -0.140203E C6 -0.137726E 06 -0.131030E 06 0I 6 -0198E06 -0.102514E 06 -0.7911E0 -0.496762E 05 -0.118285E 05 0.355629E 05 0.939395E 05 0.166076E 06 0.22.2615E 06 0.0 H19 -0.752670E 05 -0.756488E 05 -0.783513E 05 -0.817629E 05 -0. 858976E 05 -0.902924E 05 -0.9378E0 -0.974974E 05 -0.9e9738E 05 -0.981205E 05 -0.-942951E 05 -0.869155E 05 -0.154593E 05 -0.59478 0 -0.383527E 05.-0.116376E C5 0.216426E 05 0.626972E 05 __O.115757E 06 0.164136E 06 0.0 ____ 20 -0.382286E 05 -0.384237E Cs -0.39842.3E 05 -0.416467E 05 -0.438651E 05 -0.462792E 05 -004866E0 -0.505647E 05 -0.517847E 05 -0.519273E 05 -0.506445E 05 -0.475917E 05 -0.424193E 05 -0.3748k0 -0.241396E 05 __-0.100477E 05 0.779323E 04 __0.293539E 05 __0.598911E 05 0.938967E 05 0.0 21 0*193843E~ 04 0.193843E 04 0.258457E 04 0.258456E 04 Os 258457E 04 0.258457E 04 O.;547E0 0*258457E 04 0. 25 8456 E. 04 0.258457E 04 0.258457E 04 0*258456E 04 00.258M456E.04 O.;547E0 0.258457E 04 0.258457E 04 0.258457E 04 0.25845TE 04 0'.258457E 04 -:0*486306E 04 0.0 22 -0.159402E 05 -0.160391E C5 -0.212446E 05 -0.208778E 05 -0.200921E 05 -0.188664E 05 -0.1729E0 -0.152463E 05 -0.129036E C5 -0.102357E 05 -0.733035E 04 -0. 437 78 5E 04 -0*164662E 04 0.566E0 - -0.206247E 04 0.274983E C4 0.274792E 04 0.230538E 04 0.106085E 04' 0.841988E 04 0.0 23 -0.248086E 05 -0.249588E 05 -0.32.8232E 05 -0.320338E 05 -0.306350E 05 -0.287478E 05 -0.6589 0 -0.239716E 05 -0.209989E CS -0.174561E 05 -0.133068E 05 -0.879422E 04 -0.447141E 04 -0.8602E0 __ _0.1811775E 04 0.363547E (4 0.483024E 04 - 0.582175E 04 0 * 17641E 04 0.0165147E 05 0.0 24 -0.292020OE 05 -0.293514E C5 -0.382687E' 05 _-0.370698E 05.-O.3 52583 E 05 -O0.331318E 05 -O.."83E0 -0.284369E 05 -0.254752E C5 -0.216146E 05 -0.166001E 05 -O0.107241E 05 -0.524845E 04. -015346E0 0.353595E 0-3 00940382E 03 O.585448E 03 -0.470401E 03 00.2775-33E 04.0.261061E 05 0.0 25 -0.293643E 05 -0.294727E 05 -0.381314E 05 -0.367213E 05 -0.348349E 05 -'0.328590E 05 -O.30.' E0 ~-O*288352E 05 -0.260930E 05 -0.221233E 05 -0.163870E 05 -0.897955E 04 -0*200731E.04 0.1395E0 0*135566E 04. -0. 147450E 02 -0.166852E 04 _ 0.224883E 04 0.176393E 05 Q.0499934E 05 0.0 26 — 0.266108E~ 05 — 0.266729E 05 -0.343586E 05. -0.330222E 05 -0. 314089E 05 -0*299191E 05 -O.2845E0 -0.273334E 05 -0i.255464E 05 -0.227415E 05 -0.183438E 05 -0. 118901E 05 -0.404674E 04 0.157E0 kT.48553E 04 -0.154596E 04 __-0.503945E 04 -0.993001E 04 0136690E 05 0.580305E 05 0.0

27 -0.219390E 05 -0.219704E 05 -0.2-82592E 05 -0.271770E 05 -0.260119E 05 -0.251662E 05 -0.279E0 -0.245462E 05 -0.242886E C5 -0.236244E 05 -0.0221435E 05 -0. 193415E 05 ~-0*146756E 05 -w089920 -0.217990E 04 0.104445E 04 0.551476E 03 ~-0*355625E 04 0.255971E 0 5 O.*754558E 05 0. 28 -0.159101E 05 -0.159282E 05 -0.204646E 05 -0.196851E 0 5 -0.189631E OS -0.186988E 0 5 -0i028E0.-0.198592E:05 -0.210311E 05' -0.223734E 0.5 -0.236417E 05 -0.*244164E 05 -0.239926E 05 ".00Z36E0 -0.151160E 05 -0.581411E'C4 0.174822E 04 0.196837E 04 0.367004E 05 0.926055E.05.0.0 29 -0.916709E 04 -0.918605E 04 -0.117330E 05 -0.112039E 05 -0.107697E 05 -0.'108103E 05 -0.158E0 -0.128655E 05 -0.148534E 05 -0.174769E 05 -0.206947E 05 -0.242600E 05 -0.274680E 05 ~-0.277E0 -0.253508E 05 -0.138005E CS 0.206008E 04 0.142887E 05 0.484'380E 05 0.,896615E 05 0.08 30 -0.305114E 04 -0.307754E 04 -0.383036E 04 -0.349811E_04 -0.316710E 04 -0.311867E 04.-O.547E0 -0.450334E 04 -0.608132E C4 ~-0*848049E 04 -0.120195E 05 -0. 170502E 05 -0.237582E 05 -0.640E5 -0.382690E 05 -0.*370678E C5 -0.162548E 05 0.278840E 05 0.827401E 05 0.946931E 05 0.0 31 0.01 0.0 - _0.0 -- -.0 _ 0.0 0.00____ 0.0 0.0 0.0 0.0 0.0 0.0 0.0'. 0.0 0.0 __ _.0 O,00 __000.. RADIAL VELOCITY COJIP0NENT —V JI.J 1 0.0 0.0 ~ 0.0 0.0 0.0 01.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0o 0.0 0. 0 0.0 0.0 0.0 0.0 0.0 0. o 2 0.0. 0*580980E 05 0.118038E 06 0*180370E 06 0.245641E 06 0,313909E 06 0.341E0 \10 ~~0.456505E 06 0.527607E'06 0.595182E 06__ 0.655663E 06 _ 0.704512E 06 0*736225E 06 0.767E0.0.740046E 06 0*69*334E 06. 0.622899E 06 0.532254E 06 0,450417E 06 0.3,70143E 06.0.0 3 0.0 0.551072E 05 0.1111727E 06 *3.170441E 06 0.231770E 06 0.295730E 06 O,,134 0 -— 0*428558E 06 U.494364E C6 0.556763E 06 0*612943E 06.0.660017E 06 0.696284E 06 0.263E0 0.770528E 06 0.792753E 06 0.7726E0.04808E 06 0.584055E 06 0*385621E 06 0. 4 0.0 0.507899E0 0.0844E 06 0,156727E 06 0. 212884E 06 O0.27-1262E 06 0.314E0 0,3916711E 06 0.450886E 06 0*506893E 06 0.557544E 06 0.600915E 06 0.636013E 06 0.634E0 0.685231E 06 0.695978E 06 0.684144E 06 0.632312E 06 0*520394E 06 0.343039E 06 0.0 5 0.0 0.4.54628E C0- 0.919-95E 0O5 0. 140112tE066- 0. 190160E.06 0*242019E 06 0.9578 0 ___.348259E 06 0*400123E 06 0.449008E 06 0.493215E 06 0*531218E 06 0,561885E 06 05448E0 0.597606E 06 0.596811E 06 0.576109E 06 0*525721E 06 0.432455E 06 0,292583E 06 0.0 6 0.0 D.393049E Cs 0.795144E 05 0.121071E 06 0.164230E 06 mo208830E 06 O.243E0 0.299660E 06 0.343698E'C6 0.385031E 06 0.422267E. 06 0:454040E 06 0.478984E 06 0.454E0 0.501562E 06.0*493639E.06 0.467473E 06 0*418374E 06 0.3421710E 06 0.241974E 06 0.0 7 0.0 0*32.4664E 05' 0.656883E 05 0.100032E 06 -0.135675E. 06 0,172450E'06 0.293E0 0.24702.6E 06 0.282988E 0-6 0.316598E 06 0*346697E 06 0.372027E -06 0,391076E 06 0.411E0 0*.402232E 06 0.389489E 06 0.360569E 06 0.313055E 06 0.248934E 06.0.193300E 06 0.0 8 0.0 0.250792E 05 0.507754E 05 0.773803E 05 0.105018E 06 0*133540E 06 0.122E0 0.19131-3E 06 0.219100E 06 0*245000E 06 0.268062E 06 0.287150E. 06 0. 300733E 06 0.365E0 X1302934E 06 0.287403E 06 0.257506E 06 0.209716E 06 0*141394E 06 0*304166E 05 0.0 9 0.0 0.172632E C5 0.350117E 05 0*534668E 0.5.0 727231E 05 0-0926764E- 05 0.132E0 0.133295E 06 0.152901E C6 0,17L198E, 06' 0.187463E 06 0.200737E 06 0.209617E 06 0.221E0 0.206038E 06 0.190046E 06 0.162196E 06 0.'116806E 06 0.0448642E 05 ~-0*132024E'06 0.0

10 0.0 10.913023E 04 0.186163E 05 0.286144E 05 0.392074E 05 O. 503617E 05 0.6199E0 0.735813E 05 0806E0.550 05 0.105585E 06 0.113550E 06 0.118641E 06 0.1196E0 0.113145E 06 0.99386.5E 05 0.780261E 05 0.428929E 05 -0.129807E 05 -0.142009E 06 0.0 11 0.0 0.788995E 03 0.180007E 04 0.312769E 04 0.485966E 04 0.703918E 04 0.9659E0 0.126520E 05 0.0159284E 05 0.193587E 05 0.227718E 05 0 258634E 05 0.Z80262E 05 0.2811E0 0.248135E 05 0.165972E 05 0.449105E.04 -Q.149826E 05 -00.505563E 05 -0.I21663E 06 0.0 __ 12 0.0 -0.764898E- 04 -0.l52286E 05 -0.226948E 05 -0.299531E 05 -0.368751E 05 -0.4333E0 -0.490769E 05 — 0.540396E 05 -0.580389E 05 -0.609069E 05 -0.624696E 05 -0.626230E 05 -0.6151E0 -0.601114E 05 -0.599016E 05 -0.619000E 05 _-0.663316E 05 -0.859498E 05 -0. 114918E 06 00.0 ___ 13 0.0 -0o.16C628E 05 _-0.322376E 05 — 0.'485272E 05 -0.648398E 05 -0.809563-E 05 -009655E0 -0.11124E0 0.1241I E06 -0. 13 6'2- 3 0'E' 06'" -0.145586E 0 6 -0.151976E 06 -0.154506E 06 -00.152 E0 -00145633E 06 -0.136367E C6 -0.127978E 06 -0.110899E 06 -0.125648E 06 -0.119552E 06 0.0 14 0.0. -0.243142E 0 5.-0.489550E 05 -0.739879E 05 -0.993396E 0 5 — 0.124718E 06 -00.149 E0 -0.l73464E 06 -0.195529E 06 -0.215121E 06 -0.231428E 06 -0 -.243327E 06 — 0.249274E 06 -0.2475E0 __ -0.237426E 06_ -0.220775E C6 _ -0.202301E 06 -0.184010E 06 -0. 169496E 06 -O0.l27044E 06 00.0 15 0.0 -0.322381E C5 -0.650610E 05 -0.986206E 05 -0.o132898E 06 -0.167554E 06 -0.2014E0 -0.235208E 06 -0.266368E 06 -0.294426E 06 ~-0.318313E 06 -0*336674E 06 -0.347617E 06 -0.3484E0 -0.338603E 06 _-0.317485EC6 -0.289095E 06 -0.255828E 06 -0,214306E 06 -0.132979E 06 0.0__ 16 0.0 -0.396383E 05 -0.801669E 05 -0.121862E 06 -0.164807E 06 -C0.208661E 06- -0.2525E0 -0.,295692E 0 -0.336446E 06 -0.373561E 06 -0.405673E 06 -0.431270E 06 -O0.448389E 06 -004546E0 -0.446738E 06 -0.423481E 06 -0.384910E 06 -0.331649E 06 -0.256733E 06 -0.137448E 06 0.0 17A 0.0 __-0.462824E 05 -0.938063E 05 -0.l43011E 06 -0.194.146E_06 -0.246942E 06 -0.3007E0 H -0.3537246 06 -0.404691E 06 -0.4~~~~~~~~~~~~~~51618E 06 -0.492651E 06 -0. 93 6 -0.54440 0503E0 0 -0.555472E 06 _-0.530447E 06 -0.481446E 06 -0.405686E 06 -0. 296154E_06 -0.143988E 06 0.0 __ 18 0.0 -0.519046E C5 -0.105427E 06 -0.161209E 06 -0 o219'732E 06 -0.280901E 06 -0.3435E0 -0.407443E 06 -0.469335E 06 -0.527171E 06 -0*578312E 06.-0.620101E 06 -0.649862E 06 -'0.6642E0 -0.660576E 06 -0*632498E 06 -0.573643E 06 -0,477403E 06 _-0.337847E 06 -0,159002E 06 0.0 ___ 1 9 0.0 -0.562210E 05 -0.114420E 06 -0.l75460E 06 -0.240109E 06 -0.308553E'06 0305E6 -0.453860E 06 -0.527219E 06 -0.59'7423E 06 -0.660980E 06 -Oo 714005E 06 -0.752403E 06 -0.7192E0 -0.768079E 06 -0.735771E 06 -0'.668787E 06 -0.558515E 06 -0.397027E 06 -0.l927-96E 06 0.0 20 0.00 -0.589438E 05 -0.120138E 06 -0.l84631E 06 -0.253461E_06 -0.a27121E 06 -040540E6 -0,487313E 06 -0.570927E 06 -0.653426E 06 -0.731102E 06 -0.799455E 06 -''6-0"853314E 06 -0.8686 0 -0.894412E 06 -0.869262E 06 -0.804718E 06 -0.691506E 06 -0.5l8302E 06 -0.276474E 06 0.0 21 0.0 ____-0.598984E 05 -0.122163E 0.6 -0.187933E 06 -0.258389E 06 -0.334212E 06 -0455E6 -0.501406E 06 -0.590540E 06 -Co680434E 06 -0.767769E 06 -0.848323E 06 -0.916938E 06 -0.9746E0 -0'0992490E 06 -0.983263E 06 -0.929443E 06 - -0821014E 06 -0.645659E 06'-0.368457E 06 0.0 22 _ 0.0 0.467443E 04. 0.9'35100E 04 0.l39241E 05 0.182466E 05 0.22i991E 05 O.;526E0 0.288060E 05 0.313929E 05 -0.333909E'05 0.346604E 05 0.350756E 05 0.346133E 05 0.3355E0 0.313733E 05 0.2ee956E 05 0.260754E 05 0.229291E 05 0.193269E 05 0.1497L6E 05' 0.0 23 0.0 0.228546E C4 __ 0451225E~ 04 _0.659273E 04 0.843916E 04 Q.100630E 05 0.1152E0 0.I30337E 05 0 144952E 05 0.158549E 05 0.169070E 05 0.173907E 05 0.171932E 05 001654E0 0.159137E 05 __ 0:153022EC5 0.148307E 05 0.145305E 05 0.146069E 05 0.130583E 05 0.0 24~ 0.0 Q.978 3 0.149089E 04 0.205333E 04 0 o246332E 04 0.281558E 04 00.394E0 0.380467E 04 0.448264E 04 0.51374%E 04 0.550582E 04 0.519705E 04 0.395016E 04 0.2451E0 0.187154E 04 ___0236931E 04 0.378307E 04 0.592993E 04 0. 793501E 04 -0.899-199E 03 0.0

25 0.0 -0.548076E 0.3 -O.116432E 04 -0o.184137E 04 -0.252326E 04 -0.309082E 04 -0.34475E 04 -0o.355721E 04 -0o342558EC4 -0.308300E 04 -0o.260716E 04 -0.220913E 04 -0.237271E 04 -0.355645E 04 -0.453780E 04 -0.411855E C4 -0.245991E 04 -0.157682E 03 0.341742E 04 -0.865826E 04 0.0 26 0.0 _ -0.141509E C4 -0.284495E 04 -0.424541E 04 -0.552814E 04 -0.658674E 04 -0.734017E 04 -0.771755E 04 -0.761345E 04 -o0.685280E 04 -0.518400E 04 -0.235238E 04 0.152208E 04 0.498464E 04 0.612787E 04 _0. 5343 74E C4 0.404517E 04 0.32712 5E 04..__45613E 04_.E.'+_ 0.572612E _. _._03 0.0.......__... 27 0.0 -0.200739E 04 -0.398759E 04 -0.588695E 04 -O.'762434E 04 -0.912031E 04 -0.103005E 05 -0.110678E 05 -0.112649E C5 -0.106464E 05 -0.886509E 04 -0.549556E 04 -0.167799E 03 0.681389E 04 0.127739E 05._.....0.148507E 05. 0.127010E 05 0.858729E 04 0.421363E 04.-0.916628E 04 0.0 28 0.0... -0.241257E C4 -0.478004E 04 -0.705769E 04 -0.919408E 04 -0.111552E 05 -0.128988E 05 -0.143474E 05 -0.153668E C5 -0.157467E 05 -0.151913E 05 -O.133328E 05 -0.978833E 04 -0.428612E 04 0.272959E 04 0.865678 04 0.E941495E 04 0.576475E 04 -0.107744E 04 -0.546366E 04 0.0 29 _.0.0 _.....-0.....O,.248330E C4 -0.491659E (04 -C.726901E 04 -0.953624E 04 — 0.lI7664E 05 -0.139989E 05 -----------....... ~ ~ ~ ~ ~ ~ ~ ~ ~ "' "' ~~" " " ~ ~'' ~ " " ~ ~'~~"'~ —'~ -0.162341E 05 -0.184242E C5 -0.204414E 05 -0.220243E 05 -0.227136E 05 -0.217853E 05 -0.182345E 05 -0.109512.... 05 0.339951E 03 0.113410E C5 0.115575E 05 -0,484662E 04 -0.139205E 05 0.030 0.0 __-0.183686E 04 -0.362919E 04 -0.533200E 04 -0.693180E 04 -0.852488E 04 -010.2637E 05 -0.122940E 05 -0.147379E 05 -0.176945E 05 -0.212177E 05 -0.252576E 05 -0.2 95504E -05 -0o33603E 05 -0.358219E 05 -0.349684E C5 -0.295000E 05 -0.248009E 05 -0.287082E 05 -0.185867E 05 0.0 31 0.0 0.0 o0.0o 0.0 0.0 o0.0 0.0 o 0.0 0.0 0.0 0.0oo 0.0 0.0 0.0 0.o0 __ 0. 0.0 0.0 0.0oo 0.0 0.0 H(j~~~ ~~NON DIMENSICNAL TEMPERATLjRE RISE —-T(IJ) i 0.661023E 09 0.4O66617E 09 0.472951E 09 0.481880E 09 0.492565E 09 0.504777E 09 0.518218E 09 0.532434E 09 0.546878E 09 0.560891E 0-9 0.573748E 09 0.58+4547E 09 0.591854E 09 0.588784E 09 0.534301E 09 0.468383E 09 O*40 7998E 09 0.372266E 09 0.720757E..09....0220E...;24220405E 11 0.21 2 0.1'64309E 11 0.165230E 11 0..66530E 11 0.168174E 11 0.17136E - 1 0.177416.E11 0.180037E 11 0.182574E 11 0.18....48....... l 93E 11 0.186836E 11 0.188123E i 0.,186966E 11 0.169536E 11 0.148741E 11 0.129712E 11 0.117627E Ii 0.126854E 11 0.136863E 11 0.242045E 11 3 0.165675E ii 0.167038E 11 0.169105E 11 0.171800E 11 070 11..178759E 11 --- O.182792E 1 0.186979E 11 0.191123E 11 0.195019E 11 0o.198468E 11 0.201290E l 0.203305E 11 O.4284E 0.204139E 11 0.1S9698E 11 0.190766E 11 0.180741E 11i.179770E 11 0.163518E 11i 0.242045 11 4 0.166351E 11 0.168033E 11 0.170725E 11 0,174294E i 0.,178611E 11 0,183509E 11 0,188767E 11 0.194119E 11 0.199271E 11 0.203936E 11 0.207878E 11 0.210940E 11 0.213063E 11 0.214278E 11 0.214683E 11 0.214185E ii 0. 211081E 11 0..204905E.-11 0.199998E 11 0.179135E 0.24-;5 0.b1668I4E 11 0.168746E 11 0.171969E 11 0.176287E 11 0.181507E 11 0.187385E 11 0.193612E 11 0...... __,_199817E 11 0.205610E 11 0.210629E 11 0.214610E 11 0.217437E 11 0.219163E 11 0.219995E 11 0.22026E 0.2200321 0.218596E 11 0.214659E 11 0.209588E 11 0.188709E 11 0.242045E LI 6.0.167174E 11 0....169306E ii 0.172982E 11 0.177939E 11 0.183921.gE'11 0.190605E.11 0.197591E 11 0.204409E 11 0,2_"010571E 11 0.215649E.11 0.219358E 11 0.221625E 11 0.222617E 11 0.222696E 11 0.222343E Li 0.22198LE 11 0.221141E 11 0.21869E 11 0.214254E 11 0194856E L1 0.242045E 7 0.167475E 11 0.16S767E 1i 0.173821E 11 0.179314E 11 0.185925E 11 0.193260E 11 0.200831E 11 0_208074E 11.0.214413E.11 0.219356E 11 0.222605E 11 0.224138E II 0.224222E 11 0.223363E 1l 0.222233E.11 0.221548E 11 0.221117E 11 0.219660E 11 0.216300E 11 0.198911E 1L 0.2245E

8 0.i67740E I11 0.170156E 11 0. 174516E 1 1 0.180442E 11 0.187556E 11 0.19541E11 0.2031E1 0.210936E II 0.217323E 11 0.222026E 11 0.224743E 11 0.225497E 11 0.224646E 11 0.222808E 11 0.220804E 11 0.219706E 11 0.219747E 11 0.219394E 11 00216845E 11 0.199997E 11 0.242045E 9 0.167980E 11 0.170489E 11 0.175083E 11 0.181337E 11 0.188832E 11 O.197055E 11 0.20537Th 11 0.213082E 11 0.219447E 11 0.223886E 11 0.226092E 11 0.226136E 11 0.224445E 11 0.221;708E 1 0.218811E 11 0.217093E 11 0.217844E 11 0.218681E 11 0.216713E 11 0.242045E 11 0.242045E 11 10 0.L68206E 11 01170776E 11 O.175529E 11 0.182008E 11 0.189762E 11 0.198240.E 11 0.206768E 11 0.214577E 11 0.22C897E 11 0.225110E 11 0.226911E 11 0.226403E 11 0.224066E 11 0*220621E; 11 0.216931E 11 0.213992E 11 0.216004E 11 0.218065E 11 0.216970E 11 0.242045E 11 0.242045E 11 11 0.168423E 11 0.171026E 11 0.175860E 11 0.182458E 11 _ 0.190349E 11 0.198965E 11 0.202603E 11 0.215466E 1 0.221755E 11 0.225832E 11 0.227391E 11 -.226558E 11 0.2-23843E 11 0.219989E 11 0.215826E 11 0.210996E 11 0.215071E 11 0.217721E 11 0.218353E 11 0.242045E 11 0.242045E 11 12 0.168638E 11 0.171242E 11 0.176081E 11 0.182687E 11 0.190594E 11.199230E 11.207891E 11 0.215772E 11 0.222067E 11 0.226130E 11 -.227653E 11 0.226765E 31 0.223992E 11 0.220112E 11 0.216067E 11 0.213311E 11 0.215099E 11 0.217764E 11 0.220448E 11 0.242045E 11 0.242045E 11 13 0.168854E 11 0.171407E 11 0.176155E 11 0.182656E 11 0.190464E 11 0.199029E 11 0.201668E 11 0.215589E 11 0.221990E 11 0.226216E 11 0.227940E 11 0.227269E 11 0.224728E 11 0.22127E 11 0.217498E 11 0.215227E 11 0.215971E 11 0.218595E 11 0.223091E 11 0.242045E 11 0.242045E 11 14 0.169078E U1 0.171520E 11 0.176080E 11 0.182355E 11 0.189943E 11 0.198341E 11 O.209212E 1 0.214903E 11 0.221527E 11 0.226115E 11 0.228294E 11 0.228115E 11 0.226060E 11 0. 290E1 0.219679E 1 1 0.217526E 11 0.217846E 11 0.220475E 11 0. 226102E 11 0.242045E 11 0.4245 1 H 15 0.169315E 1 1 0.171589E 11 0.175857E 11 0.181776E IL1 0.1I89007E 11 0.197121E 1 1 0.2054EZ 0.213617E 11 0.22055Th 11 0.225691E 11 0.228581E 11 0.229179E 11 0.227875E 11 0.225403E 11 0.222689E 11 0.220779E 11 0.220873E 11 0.223409E 11 0.229263E 11 0.242045E 11 0.242045E 11 16 0.169575E 11.0.171619E 11 0.175487E 11 0.180907E 11 0.187622E 11 0.195297E 11 0.203472E 11 0.211552E 11 O.'218848E 11 C.224679E 11 0.228528E 11 0.230208E 11 0.229953E 11 0.228393E 11 0.226392E 11 0.224896E 11 0.224897E 11- 0.227117E 11 0.232360E 11 0.242045E 11 0.242045E 11 17 0.169870E 11 0.171624E 11 0.174977E 11 0.179737E 11 0. 185741E 11 0.192764E 11 0.200479E 11 0.208424E 11 0.216013E 11 0.22260Th 113 0.227627E 11 0.230714E 11 0.231871E 11 0.231520E 11 0.230415E 11 0.229441E 11 0.229433E 11 0.231145E 11 0.235224E 11 0.242045E 11 0.2424E1 18 0.110223E 11 0.171626E 11 0.174342E 11 0.178260E 11 0.183309E 11 0.189386E 11 0.1961E1 0.203794E 11 0.211416E 11 0.218649E- 11 0.224921E 11 0.229724E 11 0.232766E 11 0.2340E1 0.234207E 11 0.233859E 11 0.233892E 11 0.235027E 11 0.237735E 11 0.242045E 11 0.242045E 11 19 0.170677E 11 0.171668E 11 0.173621E 11 0.176487E 11 0.180275E 11 0.184988E 11 0.190595E 11 0.196998E 11 0.204006E 11 0.211312E 11 0.218503E 11 0.225079E 11 0.230534E 11 0.23445Th 11 0.236695E 11 0.237532E L1 0.237726E 11 0.238346E 11 0.239787E 11 0.242045E 11 0.242045E 11 20 0..171340E 11 0.17186*E 11 0.172915E 11 0.174490E 11 0.176631E 11 0.179395E 11 0.18289E 11 0.18704Th 11 0.192026E 11 0.197779E 11 0.204245E 11 0.211283E 11 0.218650E 11 0.225963E 11 0.2-32653E 11 0.2378E 11. 0.240409E 11 0.240686E 11 0.241224E 11 0.242045E 11 0.242045E 11 21 0.172645E 11 0.172645E 11 0.172645E 11 0.1726'45E 11 0 *172 645E 11 0.172645E 11 0.1724E1 0.172645E 11 0.172645E 11 0.172645E 11 0.12645E 11 0.172645E 11 0.172645E 11 0.17 5 0.172645E 11 0.172645E 1i — 0.172645E 11 0.172645E 11 0.172645E 11 0.172645E 11 0.22045E 11 22 0.524194E 11 0.524321E 11 0.525597E II 0.528040E 11 0.531766E 11 0.536431E 11 0.5415E 11 0.546561E 11 0.550979E 11 0.554163E 11 0.555424E 11 0.554259E 11.550748E 11 0.544148E 11 0.529387E 11 0.512971E 11 0.499698E 11 0.492551E 11 0.499503E 11 0.457376E 11 0.627079E 12

23 0*524838E 11. 0.525239E 11_ 0.528017E 11 0.533697E 11 0.542231E 11 0,552655E UL 0.637 1 0.575785E- 11 0.587358E. 11 0.597806E 11 0.605382E 11 0.607638E 11, 0.602887E 11 0.524*1 O0.579581E 1.1 0.56583~2E 11 0.5.529-50E 11 0.542200E UL 0.535507E U1 0. 533541E 11I 0.666E1 24 0.524863E 11 0.525649E 11 0. 529-955E 11 0.538544E 11 O.5,508661E 11 0.565287E 11 O.509E1 0.597250E 1-1 0-.614751E 11 0.632770E 11 0.649181E 11 0.658951E 11 0.654815E 11 O.327E1 0.618248E 11 0.600371E 11 0*586812E. 11 _0.577409E 11 0.566563E 11 0.777804E 11 O.00 4E1 25 0.524882E ii 0.526128E 11 0.531619E 11 0.542179E 11 0.556464E 11 0.572429E 11 0.593E1 0.607687E 11 0.628220E.11 0.651103E 11 0.674853E 11 O.693341E 11 0.689915E 1L 00.535E1 0.619521E 11 0.596992E 11 0*583481E 11 0.575148E, Ii 0.574693E 11 0.185211E 12 0.608E1 26 0.524892E 11 0.526541E 11 _ 0.532594E 11 0-.543754E 11 0.55801 6E 11 0.573337E 11 0.8956 1 0.6069.35E 11 0.627098E 11 C.650407E 11 0.676514E 11 0.103075E 11 0.722931E 11 0.765E1 0.666151E 11 0.638L168E 11 0.637058E 11 0.652118E 11 0.598762E 11 0.183505E 12 0.655E1 27 0.524892E 11 0.526674E 11 _0.532577E 11 0.543145E 11 0.556091E 11 0.569478E 11 0.53* E1 0.598126E 11 0.615512E U1 0.6-36098E ii 0.660255E 11 0.687762E 11 0.717285E 11 0.780E1 0.745533E 11 0.726387E 11 0.705033E 11 0,703622E 11 0.621772E 11 0.249747E 12 0.593E1 -28 0.524880E 11 0.526-507E 11 0.531647E 11 0.540734E 11 0.551532E 11 0.562239E I 1 O0.528E1 0.584282E 11 0.598023E I11 0.615197E 11 - 0.637013E 11 0.664805E iL C.699825E 11 0.726E1 0.790552E I11 0.818495E 1 1 0.800908E 1.1 0.714857E 11 0.702299E 11 0.278469E 12 0.675E1 29 0.524857E 11 0..526035E 11 O.529838E 11 0.536657E 11 0.544 680E 11 0.552258E 11 0.59581 0.566785E 11 0.576276E 11 0.589216E 11 - 0.607450E 11 0.633640E 11 0.671743E 11 0.225h1 0.806019E 11 0.905571E 11 0.888037E 11 0.832229E 11 0. 104113E 12 0.344073E 12 0O'4s 6 1 H30 0.524825E 11 0.525324E 11 C.527252E 11 0,53100SE 11 0_ O535634E 11__ C.539804E 11 0.436h1H ~~~~~~0.546163E I11 0.550264E 11 0.556539E ii 0.566487E 11 0.582675E 1 1 OQ101E1 0.540E1 0.748008E 11 0.917672E 11 0.121424E 12 0.148368E 12 0.202377E 2 C409E 12 0.685E1 31 0.524807E 11 0.524783E 11 0.524789E 11 0.524808E 11 0.524843E 11 0.524894E 11 0*545E1 0,525035E 11 0.5259.135E 11 0.525277E 11 0.525489E 11 0,525828E 11 0, 52 639 7E 11 0.5733E1 0.5291'79E 11 0.532291E 11_ 0,536918E 11 __ 0.541513E 11 0,577342E 11 0,105846E 12 0.643E1 REAL'TEMPERATURE RISE —-RT(19JJ - 1 0.952480E-0.136-1 0618E0 0.694349E-01 0.709746E-01 0. 727341E-01 0.761E0 0.76719-3E-01 0.788006E-01 0.808197E-01 0.826724E-01 0.842285E-01 0. 8528 12E -0 1 084891 0.769884E-01 0.674901E-01 0*587891E-01 -~0.536405E-01 0.103855E 00 0.348767E 01 0.386E0 2 0*236756E 01 0*238082E 01. 0*239956E 01 0.242325E 01 0.245151E 01 0.248376E 01 0.211E0 0*255642E. 01 O0.259419E 01- 0.263074E 01 0*266416E 01 0*269215E 01 0.271070E 01 0.290E0 0*244288E 01 0.214323E C1 0.186905E 01 0.169491E 01 0*182787E 01 0.197208E 01 0.386E0 3'O.238724E 01 0.240O688E 01 0.243667E 01 0.247549E 01 0.252233E 01 0.257577E 01 0.238E0 0.269422E. 01' 0.275393E 01 0.281006E 01 0*285976E 01 0.,290042E 01 0.292945E' 01- 0.245E0 O.2941,47E 01.0.287749E 01 0,274878E 01.0.260432E 01 0, 259034E 01.0.235615E.01 03487E0 4 0.239698E 01 0.2;42i1-2-2E 0 1 0.246001E 01 0.251.143E. 01 0.257364E 01 0.264421E 01 0.719E0 0.279710E_01 __.287133E, 01 _ 0.293856E Cl 0.299535E 01 0.303947E 01 0.307006E 01 0.3876E0 0.309340E 01 0. 308623i~E- 01, 0.304151E 01 0.295252E 01 0*288181E 01 -~0.258119E 01 0.486E0 5 0*240366E 01 0.243150E Cl 0.247794E 01 0.254015E 01 0.261537E 01 0.270007E 01 0.287E0 0*287920E 01 0.296267E 01 0.303500E 01 0.309236E 01 0.313308E 01 0.315795E 01 0.369E0 0.317-313E 01 0.317051E 01 0,314979E 01 0.309306E 01 0..301999E 01 0.27L914E 01 0386E0

Oo2/+08_8._ZtE__O.L..........._0__*.__2_.43956.E.. Cl 0.2/+9253E 01 0.256395E O1 0.26501/+E O1 0o27/+6/+6E Ol 0.28/+?12E 01 0'29/+536E 01 0.303/+i6E O1 0.310733E OL..............[)-;-3'L'6'O-TbE —OL...................-0;3193/+/+E O1 0.320772E O1 0.320887E 01 0.320378E 01 0.3Lc~857E 01 0.3186/+bE Ol 0.3L/+939tF. Ol 0.308722E 01[ (}.280772E OL 0.3/+8767E O1.... 7................O,._2_/t.!..3._!_SE__ 0__[.................. Q?_2_4. 46_2.0E O1 0. 250/+b2E. 0.!........ O. 258377E O!.............~.......0. -_26.79..O_3__E__~_L_..............._0~..2_78~.7_:2_.E.._0.1_................_0.._*._.2_8_:9:3.8..!._E_ 0 L O.299a/aE OL 0.308952E GL 0.3/6073E OL 0.320755E Ol 0.32296/+E OL 0.323085E OL 0.32/8/+8E O1......................... _0,3.202.20E.....0_!....................._0.-.3_..!_9.2..3.2E.01 O.3LSbLLE OZ 0.316513E 01 0.3LL671E OZ 0.28661/+E OL 0.3/+8767E 01 8 0o2,~1699E OL 0.245180E Ol t;.25L464E OL 0~260002E OL 0.270253E O1 0~281556E O1 0.293097E O! 0.3039/+1E OL Oo3L3L/+SE O1 0.3L9922E OL 0.323836EOL 0.324922E OL 0~323697E01 O.32LO/+SE O1............................. O~,.3_!_8161E O!............._0,_3_!6578EOi Oo3Lbb37E OL O~316LZSE OL 0.312/+56E 01 0.288179E OL 0~3/+8767E O1..................0..72/+204b~ OL..... 0.2/+5660E OL C.2~2279E Ol 0.261292E OL 0.272092E Ol 0.2839/+0E OL 0.295932~ O1............ 0.307033E OL 0.316205E OZ 0.32260LE OL 0.325780E01 0~3258~3E Ol............ 0.323/+07E O/.......0.3i9"463E 0i............................O_...:3.__!.5.g..89E_..q~!........................0.,...3..!__2813. E.0!........... 0.313895E O1................ 0...3.....!..5!01E O! 0.312265E OZ 0.3/+8767E 01 0.3/+8767E Ol L O......0.2/+237LE OL 0.246075E CL 0.252923E 01 0.262259E 01 0.273432E O1 0.2856/+8E OL 0.297936E 01 0.309188E OL O.3L8295E OL 0.~2436bE OL O.B26960E O1 0.326228E OL' 0.322860E OL........0.3L789?E 0],........................... 0.3!2580E OL O.308B/+SE O1 O.3LI2/+4E OL 0.3L4.'ZL/+E OL 0.312635E 01 0.3/+8767E OL 0.3/+8767E O! II o,_2~268~E_ OL..............0.2/+t~.3/+E C:L 0.253400E Ol G.26290bE O1 0.27/+278E 03. 0.28bb92E OL 0.299139E 01 0.310/+69E O1 0.31953iE O1 0.325405E O1 0.327652E01 0.326451EO1 0.3225~39E —6L............O'TB'i-6~8-6-F:-6-L —............ 0o310987EOL 0.30qO27EO1 G.309900E O1 0.313718E Oi 0.31/+629E O1 0.3/+8767E 01 0.348767E O1!2 0.2/+2993E 0l 0.2467/+6E Cl 0.253718E OL 0.263237EO1 0.27/+bBOE OL 0.287074E O1 0.29955/+E OL O.3LO910E OL 0.319.981EOL 0.325835E OL 0.328029E OL 0.3267/+9E OL...............0.32'275'5E O1................0.-31716/+E O L............................ O_,3.!..L..3.__3_.SE......O.L..................0.307363E O1 Oo3U9940E 01 0.313780E 01 0.3176/+8E Ol O.3/+8767E 01 0.348767~ O! 13 0.24~305E OL 0.246983E 01 0.253825E 01 O.263192E01 0.274443E01 0.28678/+E01 0.299232E 01 k~ 0.310646E Oi 0.319~70E O1 0~$25959E 01 0.3284/+3E 01 0.327/+76E 0i................0.3238'L/+E' Ol............0.'3'18625E oz ~0.313397E Ol 0 31012.5EO1 0.3/1197E O! 0 3l/+977EO1 0 32l/+56E Ol 0 3/+8767E OL 0 348767E OL ~...................:.....................................~...............1.4...........0...2..,3.627E 01...... 0.2471/+?E C1 0.253717E O1' 0.262759E Ol O.273692E Oi 0.285792E O1 0.2981/+3E 01 0.309658E OL O.3Lc~ZOBE OL 0.3258/3E OL 0.328952E Oi 0.328695E O1 0.32573/+E OL O.3212LOE O1 0.3Lb539E OL 0.3134.37E OL 0.3L3898E 01 0.31768bE 01 0o325795E 0.1. 0~348767E 01 0.3/+8767E 01 15 0.2/+B96~E OL 0.247245E C1 0.253395E gL 0.26192~EOL 0.2723~/+E OL 0.28~035E OL' 0.296186E OL 0.307805E 01 0.3.1'7804E01 0o3'2.5202E 0l 0.329366E 01 0.3~0228E Oi.......0',3283/+eE....0i...................O;'32'2t'788E"'b'i~'......................... 0;320876E O1 0.31812/+E 01 0.318260E O1 O.32LeL4E OL 0.3303/+9E OL O.3~a767E OL 0.3/+8767E OL............................ Lb 0.2/+~3/+",~ OL 0.2*728~E OZ 0.252863E Ol 0.260672E 01 0.2703~8E 01 0.28L/+ObE OL 0.293186E 01 0.30/+829E Ol 0.315~,3E01 O. 3237/+/+E O1 0.329290E 01 0 3'3t'7-10E-'o'i-..................'0' 3'31';~''~'E'0i................"-0';'3-2-90'9&E'-'0'i-' 0.326212E 01 0.324057E G1 0.32/+058E O1 0.327257E O1 0.33/+8i1E O1 0.3/+8767E 01 0.3/+87&TE 01..............................................................................................................................................!..7......................0...:2.~.4.76.9E. O[.......... 0.247296E O1..... 0.252128E 01................0.:25_8.9.86E 01 0.26763.7E O1 0.277758E O[ 0.28887/+E O1 0.300322E OL 0.3LL258E G L G.320759E O! 0.327992E O1 0.332/+$9E O1 0.33~LO6E OL 0.33360LE OL 0.33200~E O L 0.3.30605E 01 0.33059~E 01 0.333060E O1 0.338938E 01 0.3/+8'767E OL 0.348767E 01............................................................................................................................................................ ].8 0.2z, 5277E 01 0.2/+7298E OL 0.25L213E 01 0.256858E O1 0.264133E 01 0.272889E 01 0.282868E 01 0.293650E 01 0.304633EOL 0.315055EO1 G.324092E O! 0.3310L3E OL 0.335396EO! 0.33732/+E O1........................:0..,_..3.37./+._73..E......O.L...................O..,.3.369__72E_...._OL................ O,33.7Q2Q..E....0.1..................0.3.3865/+E OL 0.342556E 01 0.3/+8767E OL 0.348767E OL 19 0.245931E OL 0.247360E O1 0.2501i.3E O1 0.254303E.01..........0,25.976.2E_..0'i........................0o2_.6_6__55_2E.._0_.!...................__O_,...2,_7..~b.3_!E_._O_.!...... o.z~3a~sE o~. O.Zg~9~E o~ 0.30/+/+8/+E O1 O.~].~+a/+/+E 0]. 0.3~ZIE 0]. 0.3~Z~80E Ot 0.3~78~/+E ol 0..3/+ LO63E.O.L O. 342263E 01 0 ~ 342:~4Z+E 01 0.343_./-t.._3 7_E _ O!......................0__-_3_4_5.._.5.~__~___E___0_1......................_0_. 3_~..8 _?__6_7___E__0.~...................0__-.__3~.8_..7...6._7__E_.0.~1.... 2Q..............0.2/+68.Sb.E......Q~.....................Q.2.4.70/+LE__C}.l...............0,.249!.57E 0.1...... 0.251/+26E O1 0.25/+510E O1 0.258/+94E 01 0.2634'70E O1 0.269520E OL 0.276694E 01 0.28/+984E 01 0.29/+301E O1 0.304/+42E Ol 0.3L5057E O1 0.325595E 01 0.335233E OL 0.342762E O1 0.3/+6/+10E O1 0.,37_6.80__9__E__..0_.!...................._0.,.3/+7583E 01 _0_,_3_/+_.8_7.6._7E_._0..1.........._O.,..3./+__8.7.6.7_E..9L

2 1 0.248'767h0.286E 01 0.248'767E 01 0.248767E 01 0.248767E 01 0.248767E_01 0*24876'7E 010i 0.286h0.248767E 01 0.248'767E 01 0.248767E 01 0.248767E 01 0.248767E 01 0.248767E011O. 0.248767E 01 O0477E0.248767E 01C.248767E 01 0.248767E 01 0.248767E 01 0.348767E 01 22 0.755-321E 01 0.755504E 01 0. 757342E 01 0.'760862E 01 - 0.766231E_01 0.772954E 01 0.705E1 O.787549E 01 0.793915E 01.1798 503E 01 0.800321E 01 0.798641E 01 0.793583E 01 0.742E0 __0.762804E 01 O.739148E GI 0.720024E 01 0.709725E 01 0.719'743E 01 0.659041E 01 0.9059E0 23 0'.756248E 01 0.756827E CI 0.760829E 01 0.769013E 01 0.781310E 01 0.796330E 01 0.8123E0 0.829659E 01 0.846334E 01 0.861390E 01 0.872306E 01 0.875556E 01 0.868710E 01 0.8313E0 0.835128E 01 0.815317E 01 0.796755E 01 0.781,266E 01 0.771622E 01 0.768788E 01 _.9296E0 24 __0.756284E 01 0.757417E 01 0.763621E 01 0.775997E 01 0.793752E 01 0.814532E 01 0.8684E0 0.860588E 01 0.885806E Cl C.911770E 01 0.935416E 01 0.949495E 01 0.943535E 01 0.9127E0 0.890844E 01 0.865085E 01 0.845547E 01 0.831999E 01 0.816371E 0 1 0.112075E 02 0.9097E0 25 0.756312E 01 0.758107E 0 1 0.766019E 01 C.781236E 01 0.801819E 01 0.824823E 01 0.8419E0 0.875627E 01 0.905213E Cl rC.938186E 01 0.972407E 01 0.999047E 01 0.994110E 01 0916E0 0.892679E 01 0.860217E 01 0.840748E 01 0.82874LE 01 0. 828085E 01 0.266873E 02 0.235E0 26 0.756327E 01 0.758702E 01 0.767424E 01 -0.783505E 01 0.d804054E 01 0.826131E 01 0.8491E0 0.874543E 01 0.903597E 01 0.937183E 01 0.974802E 01 0.101307E 02 0.104168E 02 0.1071E0 0.959869E 01 0.91954Th 01 C.1l7949E 01 0.9-39648E 01 0.862'766E 01 0.264415E 02 0.9247E0 27 0.756326E 01 0.758894E 01 0.767399E 01 0.782628E 01 0.801281E 01 0.820570E 01 0.4017 0 0.861850E 01 0.886902E 01 0.9165b4E 01 0.951373E 01 0.991008E 01 0. 103-355E 02 O..001E0 0.107425E 02 0.104666E 02 0.101589E 02 0. 101386E 02 0______1 0 3965 2 0 H28 0.7563C9E 01 0.758653E Cl 0.7b606OE 01 0.779153E 01 0.794713E 01 C.-810140E 01 0.8530E0 H- 0.*341903E 01 0.861702E 01 0.886448E 01 0.917883E 01 0.957929E 01 0.100839E 02 0.001E2 \J1 ~~~~~~0.113912E 02 0.117938E 02' 0.115404E 02 0.111651E 02 0.101195E 02 0.401252E 02 O.940, E0 29 0.756276E 01 __ 0.757974E Cl -.0.763453E 01 0.773279E 01 0.784840E 01 0.795759E 01 0.854E0 0.816691E 01 0.830366E 01 0.849012E 01 0.875286E 01 0 9323E 01 0.967926E 01 0.149E0 0.116141E 02 0.130485E 02 0.127959E 02 0.119917E 02 0.150018E 02 C.495781E 02 O.95, 0E0 30 0.75623C)E 01 0,756948E Cl 0.759727E 01 0.765134E 01 0. 771805E 01 0,777814E 01 0.721E0 O.786976E 01 0.792885E 01 0.80.1926E 01 0.816262E 01 0. 8395b7E 01 0.878987E 01 0.9870E0 0.107782E 02 0*132229E 02 -0.174962E 02 _ 0.213766E 02 0.291608E 02 0.591776E 02 0.9693E0 31 0.756204E 01 0.756169E 01 0.7561-77E 01 0.756206E 01 0.756255E 01 0.756329E 01 0.762E0 O.1756532E 01 0.566 1 0.756881E 01 0.7571.86E 01 0.'75767.5E 01 C.754E01 0593E1 0.76250-3E 01 0.766987E 01 0.773655E 01 0.780275E 0.1 0. 831902E 01 0. 152516E 02 0.984E0 STREAM iUNCTION-SF(ItJ) 1 0.0 0.0 0..0 0.0 0.0 0.0 -0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0..... 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2 0.0 -0.472812E 02 -C.192401E 03 -0.441511E 03 -0.802481E 03 -0.128298E 04 -0.189E0 -0.261654E 04 -0.345911E 04 -0.439334E 04 -0.537925E 04 -0.634936E 04 -0.718259E 04 -0.717E0 -0.707948E 04 -0.601032E C4 -0.479529E 04 -0.3'76682E 04 -0.352133E 04 -C.491221E 04 — 0.0 3 0.0 -0927-8-29 E 02 -0.377087E 03 -0.864449E 03 -0.156986E 04 -020784E 04 -0.385E0 -0.510607E 04 -0,674444E C4 -0.855941E 04 -0.104769E 05 _ -0.123812E 05 -C.140959E 05 -O*558E0 -0.158399E 05 -0.154597E C5 -0.143275E 05 -0.127896E 05 -0.1.15051E 05 -C.108253E 05 0.0

4 0.0 -O.135314E 03 -0.549429E 03 -0.125860E 04 -0.228407E 04 -0.364620E 04 -0*5358E0_ -0.741142E 04 -0*978061E 04 — 0.124029E 05 -0.151766E 05 -O. 1795614E 05 - 0.20516 5 -0.2 263E0 -0.239506E 05 -0.243264E CS -0.236257E 05 -0.2190.12E 05 -0. 194573E 05 -0*163768E 05 0.0 5 0.0 -0.113968E 03 -0.705910E 03 -0.161618E 04 -0.293142E 04 _ -0.467665E 04 -0.6867E0 -0.949087E 04 -0.125137E C5 -0.158565E 05 -0.193947E 05 -0.229512E 05 -0.262918E 05 -0.291340 -0.311615E 05 -0.-320984E C5 -0.316932E 05 -0.297740E 05 -0.263205E 05 -0.212110E 05 0.0 6 0. 0.208006E 03 _-0.843644E 03 -0.193079E 04 -0.350057E 04 -0.558170E 04 — 0.8189E0 -0.113115E 05 -0.149023E 05 -01863E0 -.30691E 05 -0.273010E 05 -0. 313049E 05 -0.3476E0 -0.373357E 05 -0.386585E C5 -0.383843E 05.-0.362011E 05 -0.319048E 05 -0.252775E_05 _ 0.0 7 0.0 -0.236824E 03 -0.960245E 03 -0.219711E 04 -0.398219E 04 -0.634702E 04 -0.93062E0 -0.128475E 05 -0.16S144E C5 -0.214034E 05 -0.261547E 05 -0.309470E 05 -0.354937E 05 -0.3946E0 -0.4241C0E 05 -0.439690E Cs -0.436936E 05 -0.411854E 05 -0.361774E 05 -0.285353E 05 0.0 8 0.0 -0.259929E ~3 -0.105371E 04 -0.241080E 04 -0.436872E 04 -0.696120E 04 _-0.1023E0 -0.140790E 05 -0.185263E CS - 0.234315E CS -0.286211E 05 -0.338557E 05 -0.388233E 05 -0.4340E5 -0.463668E 05 -0.480-36.5E CS -0. 476540E 05 -0.447498E 05 -0.390437E 05 -0. 307296E 05 _ 0.0 _9 0.0 -0.276932E C 3 -0.112266E 04 -0.256841E 04 -0.465416E 04 -C.741527E 04 -0.1083E0 -0. 149910OE 05 -.925 S -0.249344E 05 - 0.3485 5 -0.360082E 05 -0.412787E 05 -0.4582E0 -0.492204E 05 -0.509017E C5 -0.503344E 05 -0.469505E 05 -0.403129E 05 -C.294849E 05 __0.0_ 10 0.0 -0.287532E C3 -C.116574E 04 -0.266730E 04 -0.483395E 04 -0.770244E 04 -0.1128E0 -0.155723E 05 -0*204845E 05 -0.256988E 05 -0.316242E 05 -0.37.3946E 05 -0.428577E OS -0.4757E0 H - ~~~~-0,510074E 05 _-0.526347E ~5 -0.518614E~ 05 -0.480137E 05 -0.404971E 05 -0. 273134E 05 0.0 H...... ON 11 0.0 -0.291520E 03 -0.118215E 04 -0.270558E 04 — 0.490480E 04 -0.781778E 04 -0.1142E0 -0,158148iE 05 -0.20808SE 05 -0.263152E C5 -0.321393E 05 -0.380098E 05 -0.435629E 05 -0.4834E0 -0.517770E 05 ~-0.533224E 0P5- -0.523800E CS -0.481882E 05 -0.400333E 05 -0,25-3446E 05 _0.0 12 _0.0 — 0.288772E 03 -0.117140E 04 -0.268209E_04 -0.486464E 04 -0.775810E 04 -0.1130E5 -0.157128E 05 -0.2C6870E Cs -0.261761E 05 -0.319870E 05 -0.378488E 05 -0.433966E 05 -0.4810E5 -0.515758E 05 -0.530566E~ C5 -0.520051E CS -0.476345E 0.5 -0.390-554E 05 -0.235713E 05 0.0 13 0.0 -0.279259E C3 -0.113332E ~4 -0.259638E 04 -0.471257E 04 -0.752186E 04 -0.1144E0 -0.15263~L 05 -0.201142E ~5 -0.254755E CS -0.311598E 05 -0.369042E 05 -042336E0 5 -.7440 -0.504242E 05 -0.5-18806E ~5.-0.507921E 05 -0.463680E 05 -0.375365E 05 -0.217930E 05 0.0 14 0.0 -0.263054E 03 -0.106815E 04 -0.244891E 04 -0.444910E 04 -C.7109:35E 04 -0*1046E0 -0.144639E 05 -0*19C874E ~5 -0.242080E 05 -0.296490E 05 -0.351625E 05 -0. 4041E ~ 05.4497E0 -0.482838E 05 -G.491463E 05 -C.486885E 05 -0*443031E 05 -0.354106E 05 -Q.199116E 05 0.0 15 - 0.4 - -0.240352E ~ 3 -0.976617E 03 -0.224107E 04 -0.407631E 04 -0.652300E 04 -0.9609E4 -:-0.133165E 0S -0.176057E CS -0.223694E CS -0.214455E 05 -0.326059E 05 -0.375454E 05 -0.4185E0 -0.450610E 05 -0.465243E C5 -0.455522E 05 - -0.413146E OS -0.326388E 05 -0,179260E 05 0.0 16 0i.0 -0.211491E 03 -0.859999E 03 -0.197553E 04 -0.359831E 04 -0.S76808E 04 -0.8505E4 -0.118257E 05 — 0.156715E 05 -0.199587E 05 -0.245427E 05 -0.292193E 05 -0.337165E 05 -0.3160E0 -0.406569E 05 -0.420741E C5 -0.412342E 05 -0.373102E 05 -0.292330E 05 -0.158639E 05 0.0 17 0.0 -0*176982E 03 -0. 720259E 03 -0.165642E 04 -0.302180E 04 -0.485357E 04 -0.7186E0 -0.100016E 05 -0.132930E ~5 -0.169801E 05 -0.209403E 05 -0.249969E 05 -0.289142E 05 -0.3236E0 -0.350264E 05' -0*363306E CS -0.356693E 05 -0.322761E 05 -0.252367E 05 -0.137243E 05 0.0.18. - 0.0 -Q~~~0.1307 53 6 E03 -0.560183E 03 -0.128981E 04 -0.235686E 04 -0.379369E 04 -0.5629E0 -0.78693E 04 -0.104873E ~5.-0.134465E 05 -0.166455E 0.5 -0. 199421E 05 -0.231430E 05 -0.2598E0 -0.281924E 05 -0,293248E (5 -0.288892E 05 -0.262499E 05 -0.206635E 05 -0.114287E 05 0.0

].9 0.0 -0-~4C838E 02 -0.~83487E 03 -0.8~3937E 03 -O.lbLlSE 04 -G.260952E 04 -0.388044E 04 -0.~44157~ 04 -0.~28070~ C4 -0.938519E C4 -0.~16757E05 -O.~40bO8E 05 -0.164027E 05 -0.185194t05 -0.201799E 05 -0.21C976E C5 -O.209195E G5 -0.191992E 05 -0.153817E Ob -0.877068E 04 0.0 20 0 0 -0.417858E C2 -O.194d83E 03 -0.449583E 03 -0.82383SE 03 -0.133123E 04 -0.198427E 04 -0.219116E OZ, -0.375223h C4 -0.485630E 04 -0.607713E 04 -0.73699bE 04 -0.866813E 04 -0.987912E 04 -0.108794E 05 -0.115078EC5 -0.1155906 05 -0.101943E 05 -0.89125bE 04 -0.530082E 04 0.0 2~ O.O 0.0 O.O 0.0 0.0 C.O 0,0 0.0 0,0 G.O 0.0 0.0 0.0 0,0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 Z2 0.0 -O. iblOgOE CI -0.648283E CI -0.145343E 02 -0.255100E 02 -0.~89190E 02 -0.540869E 02 -0.702003E 02 -0.8~3095EC2 -0.1013226 C3 -0.114053E 03 -0.123389E 03 -0.128556E 03 -0.1293476 03 -O.I~CZ~IE 03 -0.12C~79E 03 -0.11369~E 03 -0.107096E 03 -0.1010506 03 -C.101239E 03 0.0 g3 0.0 -0.251484E G1 -0.10059~E 02 -0.2245741: 02 -0.391991E 02 Og -0.595081E -0.~25094E 02 -O.lO?2?lE 03 -O.132685E C3'0.157313E 03 -0.179325E 03 -0.196720E 03 -0.201952E 03 -0.212623E 03 -O.21130Ck 03 -0.2050236 C3 -0.194940E 03 -0.1817396 03 -O.IO5017E 03 -0.142596E 03 0.0 24 0.0 -O.i$~Oii- C1 -0.117~52E 02 -0.261963E 02 -0.464010E 02 -O.b87162E 02 -0.951675E 02 -0.124038E 03 -0.154272E O~ -0.1842076 03 -O.ZllBCgE 03 -0.232o73E 03 -0.245703E 03 -0.251028E 03 -0.251545E C3 -0.249568E 03 -C.Z4736ZE 03 -0.246778E 03 -0.2491b16 03 -0.23I?95E 03 0.0 25 0.0 -0.2976616 01 -0.118138E 02 -0.201165E 02 -0.4blZZSE 02 -G.68047bE 02 -0.943034E 02 -O.123299E OB -0.154031E C3 -C.IU465?E 03 -0.212126E 03 -0.232108E 03 -O.Z40560E 03 -0.239050E 03 -0.235290E 03 -0.2J~z+56E C3 -0.235350F C3 -0.2414186 03 -0.246790E 03 -0.146445E 03 0.0 gO 0.0 -0.Z6~7486 C1 -0.1007686 02 -O.ZB541OE 02 -0.4061,56 02 -0.613285E 02 -0.8538106 02 P-J -0.112496E 03 -0.142011E 03 -0.172583E 03 -0.2017576 03 -0.225071E 03 -0.239291E 03 -0.239520E 03 }.a -0.2~4Zq5k C3 -0.233481EC3 -0.241040E C3 -0.257115E 03 -0.290281E 03 -C. lb4096E 03 0.0 — q 27 O.O -0.22239IEC1 -0.8791396 O1 -O.193b?4E 02 -0.334348E 02 -0.500817E 02 -0.711376E 02 -O.949571E 02 -0.122138k 03 -0.152306E 03 -0.184475E 03 -0.2167246 03 -0.245827E 03 -0.267036E 03 -0.276405E 03 -0.2]5678E C3 -0.271941~- 03 -0.273147E OB -0.289289E OB -0.137430E 03 0.0 20 O.O -0.161278ECl -0.o372~,5E O1 -0.140315E 02 -0.242367E 02 -G.BbSg46E 02 -0.523173E 02 -O.71066BE OZ -0.937b?3E C Z -O.lZlOOSE 0.3 -0.153204E 03 -0.190300E 03 -O.231223E 0~3 -0.273009E 03 -0.3t;g129E 03 -0.332108E03 -0.333877E 03 -0.324322i::: 03 -0.324487E 03 -0.134681E 03 0.0 g9 0.0 -O.9292bOE CO -0.366853E Oi -0.805360E 01 O g -0.138499E -C,210678E 02 -0.301379E 02 -0.418282E 02 -0.571459EC2 -0.713525E 02 -0.1039706 03 -O.13859ZE 03 -0.182272E03 -0.234131E 03 -0.288783ti 03 -0.332775EC3 -0.344618E 03 -0.32383t~1:: 03 -O.gT/b4OE 03 -O.q??782E 02 0.0 30 0.0 -0.309295E CO -O.121800E 01 -0.263711E C1 -0.440Z~OE 01 -O.b47581E O1 -0.912129E 01 -0.1290086 Oi -0.186398E C2 -0.2757C9E 02 -0.416069E 02 -0.036206E 02 -C.~75596E 02 -0.147917E 03 -0.217290E 03 -0.2cidbiTEC3 -0.306720E 03 -0.36377BE 03 -0.257598E 03 -G.622091E 02 0.0 3i 0.0 0.0 0.0 0.0 0.0 C.O 0.0 0.0 0.0 0.0 0.0 0.0 0.0 O.O 0.0 0.0 C.O 0.0 0.0 0.0 0.0 RADIAL VAPOR VEL~CIIY C~1~F{JINENT AT INTERFACE —- VG{JI 0.0 0.639192E C4 0.128222E 05 0.191908E 05 0.253334E 05 0.310,94E 05 0.361499E 05 0.404735E 05 0.43~7~4E C5 0.402381E 05 0.4745756 05 O.475127E 05 0.464876E 05 0.445805E 05 0.4ZllgelE 05 0.395183E C5 0.371bO~E Ob 0.353467E 05 0.342054E 05 0.345637E 05 0.0

iN~TERFACE VtLuCITY UUL TL INTERFACE CHANGt —-UL(J) -J.3 151 5E-C1 -0.324b650E-C1 -C.344721E-C 1 -0.375375E-01 -0.418359E-01 -0.47594bE-0 -0.55097b-0 i -0.648SCE-COi -0.767604E-01 -0.917355E-01 -C.110030E 00 -0.13193b6E 00 -C.157301E O0 -0.185205E 00 -0.212i33Jt CC -0.235093E CC -G0.246334E CC -0.2463o4L 00 -0.248436E 00 -C.246463E 00 O.0 VAPuK STREAtP FUNCTIUN AT INTERFACE —-SFI(J) O.O 0./22756t- CO 0.d91G26E 00 0.200481L CI 0.356410E 01 (G.556891E 01 0.801923E 01.lCi9151E 02 G.1425o4E 0 0.180433E C2 0.222756: 02 0.269535E 02 C.320169E 02 C.376458E 02 O.436bC2L 02 -.501202E,2 0.57025bE C2 C.643766L 02 0.721731E 02 0.804151E 02 0.0 HEAT TRASFER CLEFFICI:IdT CF LIGUID —-HL(J) O.102777L-01 0.312332E-01..317658E-01 C.320641Et-01 0.322575E-01 C.324098E-01 0.325391E-01 0.326549E-01 O.32738b7E-CI 0.328460E-0l 0.329434E-01 0.33040;E-01 O. 331384E-01 0. 332403E-Cl 0.33349CL-01 U.334Ab6E-Cl 0.336055E-01 C.33770oE-01 0.i39175E-0I C.339523E-01 0.14247bE-0 HEA' TRANiSFER CCEFFiLICIENT OF VAPOk —-hG(J) F-J H..18846Z-02O 0.847101L-G5 C.83210 VE-C5 0.777129E-05 0.530393E-05 0.537205E-05 0.391993E-05 CX)D 0.338347E-05 0.233831E-C5 C.183483E-C5 0.713526E-05 NUL= 5158.742188 NUG= 1C43.777588 RMAX= 0.1 C17C70-03 RRI= 0.993754E 00 NEI= I NE2= 1 KAL= 3.133192E 11 RAG= 0.6b5264E 12 CL= 15.1.85343 CG= 1.155738 RP= 1.497664 DQILDT= -0.017565 LJi= C.62.343L C2 OQS= C.1 71359E CO DQL= 0.51C014E 02 EXECUTIOUN [ERMIthATIED

REFERENCES 1. Clark, J. A. and Barakat, H. Z. Transient Laminar Free-Convection Heat and Mass Transfer in Closed Partially Filled Liquid Containers: Technical Report 04268-6-T, Office of Research Administration, University of Michigan, Ann Arbor, Michigan, Contract NAS-8-825, Marshall Space Flight Center, January 1964. 2. Barakat, H. Z. and Clark, J-. A. Transient Natural Convection Flows in Closed Containers: Technical Report 04268-10-T, Office of Research Administration, University of Michigan, Ann Arbor, Michigan, Contract NAS8-825, Marshall Space Flight Center, August 1965. 3. Merte, H., Clark, J. A., and Barakat, H. Z. Finite Difference Solution of Stratification and Pressure Rise in Containers: Technical Report 0746130-T, Office of Research Administration, University of Michigan, Ann Arbor, Michigan, Contract.NAS-8-20228, Marshall.Space Flight Center, January 1968. 4. Coeling, K. J. Incipient Boiling of Cryogenic Liquid: Ph.D. Thesis, University of Michigan, December 1967. 5. Clark, J. A. A Review of Pressurization, Stratification and Interfacial Phenomena: International Advances in Cryogenic Engineering, Vol. 10, pp. 259-284, 1966. 6. Thomas, P. D. and Morse, F. H. Analytical Solution for the Phase Change in a Suddenly Pressurized Liquid-Vapor System: Advances in Cryogenic Engineering, Vol. 8, pp. 550-562, 1964. 7. Knuth, E. L. Evaporation and Condensation of One Component Systems: Journal of the American Rocket Society, Vol. 32, pp. 1424-1426, 1962. 8. Yang, W. J. Phase Change of One-Component System in a Container: AIChE Paper No. 63-A48, presented at the 6th National Heat Transfer Conference, Boston, Mass., August 1963. 9. Yang, W. J. and Clark, J. A. On the Application of the Source Theory on Problems Involving Phase Change-Part 2: J. Heat Transfer, Trans. ASME, Ser. C. Vol. 86, pp. 443, 1964. 10. Yang, W. J., Larsen, P. S., and Clark, J. A. Interfacial Heat and Mass Transfer in a Suddenly Pressurized, Binary Liquid-Vapol System: Trans. ASME, J. Eng. for Ind., Vol. 87, pp. 413, 1965. 119

REFERENCES (Concluded) 11. Epstein, M., Georgius, H. K., and Anderson, R. E. Generalized Propellant Tank Pressurization Analysis: International Advances in Cryogenic Engineering, Vol. 10, 1966. 12. Huntley, S. C. Temperature-Pressure-Time Relations in a Closed Cryogenic Container: Advances in Cryogenic Engineering, Vol. 3, p. 342, 1960. 13. Leibenberg, D. H. and Edescuty, F. J. Pressurization Analysis of a Large Scale Hydrogen Dewar: International Advances in Cryogenic Engineering, Vol. 10, 1966. 14. Jacob, M. Heat Transfer: Vol. 1, John Wiley and Sons, Inc. Chapter 29. 15. Swim, R. T. Temperature Distribution in Liquid and Vapor Phases of Helium in Cylindrical Dewars: Advances in Cryogenic Engineering, Vol. 5, 1960. 16. Brenteri, E. G., Giarratano, P. J., and Smith, R. V. Boiling Heat Transfer for Oxygen, Nitrogen, Hydrogen and Helium: NBS Technical Note No. 317, September 1965. 17. Evaluation of AS-203 Low Gravity Orbital Experiments: Technical Report HSM-R421-67, Contract NAS-8-4016, Space Div., Chrysler Corporation, Huntsville Operation, January 1967. 18, A Compendium of the Properties of Materials at Low Temperature-Phase I, Part 1, Properties of Fluids (WADD Tech. Report 60-56, Part 1), Nat. Bureau of Std., V. J. Johnson, General Editor, July 1960. 19. Barron, R. Cryogenic Systems: McGraw Hill, 1966. 20. Weber, L. A., Diller, D. E., Rader, H. M., and Goodwin, R. D. The Vapour Pressure of 20~K Equilibrium Hydrogen, R-236, NBS, March 1962. 21. Roder, H. M., Weber, L. A., and Goodwin, R. D. Thermodynamic and Related Properties of Parahydrogen from the Triple Point to 1000K at Pressure to 340 Atmospheres: NBS Monograph 94, NBS, 1965. 22. Personal communication from Marshall Space Flight Center. 120